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What sort of supply network would a war fleet such as that led by the Tempest need? Or for that matter, what sort of supply network would any space going war fleet need?

Resources that are going to require supply are:

Replacement Craft 
Crew Consumables (Food, Air, Water, Gear & Clothing) 
Replacement Crew 
Ship Consumables (Coolant, Lubricant, Filters, Propellant, Secondary Fuels) 
Replacement Parts (including materials for fabrication and repair)

Fuel is going to be your critical resource that determines endurance, and it's highly variable... if fuel is carefully conserved, a ship could travel dozens of systems (though not very fast) and stay out for six months or more; but under heavy acceleration and combat conditions, it could expend the same amount of fuel in just a few days. Loroi and Umiak ships would use types of matter-to-energy conversion reactors for power, so fuel in our case is an exotic, volatile material (like antimatter) that requires sophisticated containment (so that your ship doesn't blow itself to bits when you take engine damage or lose power). Ratio of actual fuel mass to containment apparatus mass is likely to be about 1:10. In WWII, large naval warships typically devoted about 15% of the ship's mass to fuel and stores; applying this figure to a 350,000 ton Loroi cruiser, we get a fuel/storage mass of 52 kilotons, or an actual fuel mass of 5.2 kt. Using a reasonable fudge factor for inefficiency in matter-to-energy conversion, this is enough fuel to run the engines at full thrust (30G) for roughly 100 hours. This can be enough fuel for a strike mission lasting about a month, assuming that the ship travels at an aggressive but reasonable pace to the target, makes several hyperspace jumps (both there and on the return voyage), and spends no more than about two continuous days of intensive combat operations. An Umiak warship of similar size would have a lower maximum thrust, more efficient engines and a higher fuel mass percentage, giving it a full thrust endurance perhaps as high as 200 hours.

Propellant, that is, reaction mass, is probably not going to be significant compared to reactor fuel (given that Loroi drives are "mostly" reactionless). Some additional propellant will probably be required for auxiliary systems (such as maneuvering thrusters), but I would class this as a normal ship consumable.

Munitions, mostly torpedoes and point-defense missiles, are a secondary and limited resource in a beam-weapon dominated battlefield. Loroi commanders typically use torpedoes like the St. Louis Rams use Time-Outs; they only get a few to start with, expend them at the first halfway-decent opportunity and then forget about them for the rest of the battle.

Most other consumables will be used at a fairly constant rate. Food, in particular, I think is going to be a very minor resource in terms of endurance; Loroi ships are very large and the crews are comparatively small. Compare a Loroi cruiser (350m, 350 kt) crew of 400 with the crew of 2,500 for a first-rate WWII battleship (260m, 50 kt). I would expect most warships (under normal circumstances) to carry food and normal consumables to last at least 6 months.

On the strategic level, both sides are going to need substantial shipping infrastructure to move materials, people and supplies all around their respective empires. Luckily, the majority of this shipping is going to be behind the front lines and therefore mostly inaccessible to enemy forces. More dangerous will be supplying bases at the front, and even more so the task of underway replenishment for squadrons in the field. All manner of civilian transports and tankers will be used for routine supply, but military transports and tankers will generally be required for underway replenishment or providing supply for long-range offensives. These will be faster and less fragile (and more expensive) than their civilian counterparts. On typical strike missions across the no-man's-land, ships normally have enough range that tankers are not necessary, unless the target is at unusual range, or if extended combat operations at the target are expected.

When on defense, most Loroi heavy units are going to stay near their bases, but underway replenishment will be important for the fast strike squadrons (such as the Tempest's) that range through the no-man's-land. With proper coordination (making use of Farseers for rudimentary FTL communication as well as detection), underway replenishment can keep these squadrons operating away from base for months on end. As long as the Loroi can keep the supply convoys away from Umiak activity, they don't even have to be escorted.

Logistics are more of a headache for the Umiak than the Loroi, not only because they are on the offensive most of the time, but also because unlike the Loroi, Umiak commanders must arrange for supply rendezvous in advance, and their convoys must be escorted, because they don't always know where the Loroi raiders are. Forward supply depots can be an important aid to extending range or supporting lengthy battles far from friendly bases, but they must also be well defended or they will be quickly destroyed by Loroi raids.

Well, these big ships can accelerate at 30 gees. It probably isn't all that expensive for them to drag the stuff up out of gravity wells.  

Right. Even for the backward Humans, I don't think the cost of lifting components into orbit is going to be that big a deal, compared to the rest of the cost of building a starship (and the colossal amount of energy required to fuel it). No doubt some exotic materials may be more efficiently fabricated in microgravity, but I would bet that most manufacturing is much easier to do planetside.

Aside from atmospheric and gravitational stresses, the biggest problem with having starships lift off from planets by themselves is that their main engines would do catastrophic ecological damage to the planet. I doubt maneuvering thrusters alone would be sufficient to lift a sizable starship into orbit, and these are probably also fairly toxic. Intermediate-power atmosphere-safe drives would be a waste on ships that will have no use for them after launch. So there are probably specialized jump-capable starships that are designed for taking off and landing on planets, but these will probably be the minority.

I'd assume that the Outsider-Terrans already have at least 1 space Elevator finished.  

So would I, and remember that Humanity has six inhabited worlds, not just one. Though the extra-solar colonies are too young to provide much industrial might, the Mars colony is over 100 years old and is a major industrial site... and has only a 1/3 G gravity well to climb out of.

The engines are the primary power source for an Outsider starship; each engine consists of a matter-conversion reactor and a reduced-reaction drive. Although most vessels will have auxiliary reactors, most heavy weapons will require main power from the engines in order to be able to fire. However, the power required to fire a ship's weapons is relatively small compared to the total produced by the engines (a cruiser's engines must produce roughly 100 terawatts to drive it at 30 G), so you probably don't have to divert more than 10-20% of your engine power to fire all your weapons. So, power is not the limiting factor on the number of weapons a ship can carry.

Rather, cost, volume, mass and required ship infrastructure of the weapon are more likely to be the limiting factor. Like battleship guns of old, a beam turret has more mechanism below the surface of the hull than the part you see above, in addition to extra power transmission and heat management infrastructure that must be added to the ship for each weapon mount. Also, aside from the engines, the main weapons are going to be the most expensive components of a starship.


Is it possible for one ship to "tow" another through a jump-point?

I don't think so. You can tow another ship into the correct position and vector for a jump, but unless the towed ship has an operating jump drive (which is unlikely if it has to be towed), it can't jump on its own, and the tug can't "pull" the towed ship into hyperspace with it with its own jump drive, unless it was specifically designed to do this. Such a supertug would essentially have to be a carrier, and one that could handle warships larger than a destroyer would have to be pretty huge. Probably easier in this case to have mobile repair facilities brought to the crippled ship.

Maintaining supply lines is mostly going to be an issue when your forces have pushed deep into enemy territory, and you have to send supply convoys through territory that is not firmly under your control.

Here's a thought, assuming for the moment that main battle unit ships can survive the average engagement (i.e., no "one-shot, one-kill"), how robust are the FTL drives? From what is said above about the difficulties of towing a ship FTL, we can assume one of the following: a) The jump drives are very robust and well protected, b) There are multiple jump drives on each ship, and each is small, replaceable, and relatively inexpensive, c) There are mobile shipyards in each system along the "Steppes" to repair ships, d) Once FTL drives are damaged, that ship will stay in the system until destroyed. Option "d" would make the Loroi strategy of a handful of top-line ships unfeasible, so that seems unlikely.

a. Jump "drives" are field generators that allow the ship to slip from normal space into hyperspace; they are relatively small and are usually situated in an interior part of the ship (not sure exactly where... checking my subsystems layout diagrams for Tempest, it seems I neglected to add the jump generators). They are not especially robust, but they are fairly well protected; if your jump drives have been damaged, this means that enemy weapons have penetrated screens and armor to the core of your hull, and so chances are you've suffered catastrophic damage to other systems as well (notably engines).

b. A large ship probably has more than one physical generator, and hopefully there's some redundancy built in. Damaged generators can often be repaired. However, the generators require a great deal of power to operate (they must be charged over time in a similar manner to the Wave-Loom device), so a loss or significant reduction in engine power (or damage to the power accumulators) may also render a ship unable to jump. Usually, if a ship can make headway in real space, then it can jump.

c. There are a limited number of mobile shipyards available at bases along front, but...

d. A Loroi raider ship that has received crippling damage while in action out in the Steppes is probably doomed. Help may be weeks away in arriving, and the Umiak conduct their own patrols through the Steppes; there's not a lot of time for extensive repairs out in the no-man's-land. Unless help is close by, or the area is relatively quiet, or perhaps there are sufficient friendly forces in the area, then if a ship receives irreparable damage while in uncontrolled territory, it will probably be abandoned and scuttled.

A jump drive is a device that generates a field (possibly a gravitational field, or an intense electromagnetic field) that alters the curvature of local space so that the drive and the vessel containing it can slip through the fabric of realspace into hyperspace. The drive must generate a field large enough to enclose the entire ship (and any ship that it is towing). The jump field is normally not much larger than the generating ship. The field disappears into hyperspace along with the jumping ship, so any "hole" in space-time that is created is only momentary.

If a starship that was towing another object at the end of a tether (outside the reach of the towing ship's jump field) activated its jump drives, the tether would snap as the towing ship disappeared into hyperspace, while the towed object continued in realspace on the same vector. If the tether were infinitely strong, I supposed it's possible that the towed object might be dragged into hyperspace along with the jumping tug, but more likely the tug's jump vector would be retarded by the drag of the towed object, and the jump would fail (either dropping back into realspace, or disastrously continuing on an unintended vector off into hyperspace, perhaps being pulled back into the local gravity well or careening off on a vector that does not return the ship safely into realspace).

So a jump-capable tug needs to have additional jump drive generators that can extend the field to cover the towed object, perhaps in the form of booms that house drive generators and extend aft to enclose the towed object; the towed object must be held fairly close to the body of the tug so that both can enter hyperspace in a single jump field "bubble". It also needs to have enough power to generate a field sufficient for both its own mass and the towed object's mass.

The jump drive doesn't open a "window" for the ship to fly through in a physical sense. The jump field is a three dimensional envelope, and whatever is inside the field boundary slips into hyperspace... it simply disappears in a flash of electromagnetic radiation. Once the jumping ship is gone, there is no residual "hole" for a towed ship to fly through. If the towed object is within the field envelope of the towing ship (for example, if the field is spherical and the generating ship is long and narrow, and the towed object is alongside the jumping ship), then both will enter hyperspace together... but this will mean that the towed object will have to be very close to the jumping ship... as in a distance of less than the length of the jumping ship, nearly touching it. If the jumping ship and the towed object are going to be this close, it would be wise to secure them together in some fashion; you really don't want a collision in hyperspace. There is also the matter of providing inertial damping for the towed object. As mentioned in the FTL thread, it seems very likely that the jump field and/or the transition to hyperspace may cause substantial spatial and/or gravitational disturbances that could damage any occupants of the towed vehicle, or even the vehicle itself, unless counteracted. Allowing for the likelihood that the towed vehicle is unable to generate artificial gravity for itself, the towing ship will need to somehow provide this damping capability.