I can only definitively answer part (1), and that answer is:
Yes, once you have explored a system you retain that information and it shows on the System Map for you.
I understand from the game forums that the first players to survey a system get more income from doing so than later players - but that later players do get some (presumably this data is confirmation of the earlier surveyors' findings).
I recall from memories of statements by the developers (can't cite, sorry) that in the released game, systems that have been explored and see a lot of player traffic will eventually have space stations built and come under faction control. I don't know whether the faction that players sell the data to has influence on the faction that moves into the system.
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."
Best Answer
It depends on the mission but generally, yes!
When you accepted a mission you may have noted that it already listed a penalty for failing it. Should you fail, this penalty will be issued as a fine (which may become a bounty if you let it expire).
In addition, missions are associated with one minor and possibly a major faction. Minor factions include, for example, the Sirius Corporation, while major factions include, for example, the Federation. Should you fail a mission, the associated factions may loose some of their influence, depending on the value of the mission. In addition you will loose some of your standing with them (reputation). Note that your rank, once attained, remains unaffected as you cannot be de-ranked.
In addition, some missions, notably those associated with Powerplay, may impact the preparation, expansion, and maintenance phases of a faction should you fail a mission associated with these. This in turn may have an impact of your ability to gain merit with the power or may, eventually, even cause your faction to drop out of Powerplay altogether should it fail its cycle goals.