Which Faction Power will result in the greatest bonus troops and/or fewest troop losses?
Die Mechaniker

1) Your starting HQ is always treated as FORTIFIED (+1 to both dice) when you defend it.

2) If your defense roll is two natural 6s, that territory cannot be attacked again for the rest of the turn.

Reason: The odds of getting two natural 6s on a single roll is 1/36. Therefore, the odds that we have not rolled a pair of 6s by the nth attack is, (1-1/36)^n. This is roughly 50% by the 25th-26th attack. This means that in half the games that you roll with two defenders 25 times, you will still not have recieved a bonus from this power. I would say on average, the first game lasts anywhere from 3-5 rounds, and considering you will probably only get 3-5 troops per round, for a total of 8+[(3*5)|(5*5)]=[21|33] troops. If you only received the minimum troops each turn, you more than likely would never get the faction bonus. Even if you are lucky enough to roll 66, the only fair way to count troops saved would be to divide the troops that you could have lost by the odds of you rolling 66 (approximately 21/25 or 33/25, somewhere close to 1 troop. If you could get the 5 troops per turn (probably only likely by getting a continent bonus from Australia/South America/Africa, you would be much better off with a FORTIFIED Headquarters.

A fortified HQ gives you +1 to your High and Low dice. This bonus will result in you losing ~0.388 troops when compared to a normal unmodified attack. This means that you only need to have 4-5 troops within your HQ before this faction power is certain to have a greater effect than the pair of natural sixes power.

The Saharan Republic

1) You can make your maneuver at any point during your turn.

2) When making your maneuver, you may maneuver between any two territories you control, even if they are not connected.

Reason: While neither of these power directly results in fewer troop loses, the ability to maneuver at any point in your turn could allow you to move troops that were previously on one of two different fronts (South America/Africa), for a final assault against two opponents HQs. I will edit this with a little more detail later on troop loses that could be avoided because of this.

Imperial Balkania

1) When recruiting troops, round up when dividing your territory and population by 3, not down.

2) You still draw a resource card at the end of a turn where you expanded into 4+ territories, even if you didn't conquer a territory.

Reason: You aren't likely to be able to get 10+ territories in a 5 player game. At least with the second faction power, you will most likely get 1-2 resource coins. This will amount to at least 2 troops over the course of a game.

Enclave of the Bear

1) The defender subtracts 1 from his lower defense die in the first territory you attack during your turn.

2) If your attack roll is a natural three of a kind, and at least one defending troop is defeated, you conquer the territory. Remove all defending troops.

Reason: From the same statistics link above (Die Mechaniker), you can see that -1 do the defenders low die will result in 0.14 less troop loses, but only for the first territory that they attack in a turn. A clever opponent though can place a 1-2 troops in front of their main defenses to mitigate this power somewhat. The odds of rolling a natural 3 of a kind that kills at least one troop is 125/7776. The number of turns before that is likely to happen is about 43 attacks with 3 troops. I don't think you are likely to see this within the first game. Even if you did, you probably would only kill at most 5 troops, saving you about 5 troops yourself (5/43=0.116). Two attacks with Enclave of the Bears first power would already save you more troops in the long run, and is much more likely to happen.

Khan Industries

1) At the start of your turn, place one troop in each territory that has an HQ you control.

2) When drawing a territory card, you may place one troop into that territory if you control it.

Reason: You are certain to get 1 troop as long as you control your HQ. This will likely gain you 3-5 troops before the first game ends. The second power only gets you a troop if you defeat an enemy, and if you have that territory under your control at the end of the turn. You might have to waste a maneuver to bring those troops to the front lines, but you will earn more troops with the first power.

Overall, I think that Khan Industries is likely to earn/save you the most troops in the first game. With Die Mechankier placing a close second, if you can place your HQ at a good choke point.