You are asking multiple questions which would, together, take quite a bit to explain. I will try to sum up a quick answer and would like to direct your attention to the manual as PDF, specifically the Powerplay section (PDF page 75, manual page 112) to get in-depth answers to your question.
How do the major factions (federation, empire, alliance) expand?
They expand by having their member powers expand. Powers expand in three phases, each which takes at least a cycle (typically a week) to complete: Preparation, Expansion, Control.
In the preparation phase a potential system must meet specific requirements (population, no other faction direct control or exploitation) and the power must have enough resources to expand. Then a potential system must make it to the top 10 of prepared systems to start an expansion attempt. During the attempt several criteria must be met. If the attempt is successful, the system becomes, in the following cycle, controlled.
If a given power [...] conquers a rival system does it count for their faction?
I do not believe it is possible to directly conquer a rival system. That system would need to loose its current power's control or exploitation via the turmoil and revolt mechanics (basically the reverse of the above) to become available for expansion (as above). However, it might be possible during a war scenario but I haven't seen it happen (yet).
Do combat zones outside the realm of the 8 powers affect the territory of the 3 major factions?
They might as the combat zones might be linked to a power's expansion attempt. Check the expansion attempts for the powers to be sure.
Can powers from the same faction be each other's enemies?
Yes and no. They are rivals but not so much enemies. Rival powers can, for example undermine or oppose expansion attempts. To quote the manual, page 112: "[...] Ultimately, every power stands alone."
After sciencing with @Yuuki, yes. PowerPlay merits are not only shared within Wings, but you don't even have to be in the same power. Normal rules apply, though. (ship must be damaged by everyone to share the rewards)
Best Answer
I believe the best way to make 100,000 Cr quickly while helping your Power, is to find a trade route between systems your Power needs deliveries to/from, and combine the PowerPlay deliveries with trading.
I believe all of the powers have Preparation, Expansion and/or Control PowerPlay actions they ask of you that are basically deliveries. Some are between headquarters and other systems. Some are between a control system and a preparation or expansion target system. You need to read the description for your particular power to figure out where the deliveries need to start and end.
Figure out a few different PowerPlay delivery routes that will help your Power. Check out what kind of economies are at each station (or use one of the trading tools). You should be able to figure out some trading options that line up with PowerPlay deliveries. Fill your cargo or multi-purpose ship up with a mix of cargo and PowerPlay deliveries. Instead of spending your money on more PowerPlay things to deliver when you haven't reached the timeout to pick up more, just do runs of cargo until that timer resets.
If you stick with purely powerplay things, you need to get up in the ranks to really make money, and there's a delay 1-2 weeks before you start getting that salary. Rank 5 earns a 50,000,000 Cr a week salary, but requires 10,000 merits. Merits partially carry over from week to week, so once you get to rank 5 you may only need to earn ~3,500 merits a week to stay there... More reasonably, Rank 4 only takes 1,500 merits and earns 5,000,000 credits a week. 1,500 merits is just 100 interdiction kills of rival power npcs in a hostile system...