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."
You get merit points by participating in pretty much anything Powerplay related. Expansion, Fortify, Prep, Undermine, etc.
The easiest and most painless way to do it is by Fortifying (although it is very time consuming). You do this by collecting fortification materials from HQ and delivering them to a control system that you own. Preparation is a similar action, just taking the data to a new uncontrolled system from any controlled system. (1 merit per item)
Expansion for Arissa has you enter the prepared system and enter a conflict zone like area. Other powers have different objective, but it's still the same rewards. The merits aren't much (1 per objective) but a decent amount of time and hanging around allies could get reasonable merits pretty quickly.
Undermining and Opposition is by far the best way to get merits, but it is also the most dangerous. To undermine powers not aligned to your major faction, you must go into their controlled system and kill specific transport ships, either by interdicting them or hanging out in the nav beacon. Undermining and opposing an ally major faction has you pirating these transport ships instead of killing them, try and do this as non-combat as possible, they are allies after all. Be warned, this will be an illegal act and attract bounties on your head as well as the security force coming down on you. (15 merit per kill/loot)
If you attack any ship that is in a power aligned to your major faction, you will lose merits, so always avoid lethal engagements with allies. (-10 merit per ally kill))
Best Answer
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)