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Combat Intelligence affects how many AP a soldier will get when they rank-up.
A "standard" soldier gets 3 AP per rank, times 7 ranks, equals 21 extra soldier AP at max rank.
Above average gets 1 bonus on top of that, so 4 (3+1) per rank times 7 = 28 at max.
Gifted gets 2 bonus, so 3+2 times 7 = 35 at max rank.
Genius is 4 bonus; 3+4 times 7 = 49 at max rank.
Savant gets 7 bonus, 3+7 times 7 = 70 bonus AP at max rank.
In general combat intelligence isn't super important. You'll be spending a lot more shared XCOM AP than anything else, and you should have more than enough to go around as long as you save it for your better soldiers.
However, what's great is that those extra personal AP points can be added retroactively if you raise their combat intelligence through covert ops. So if you send a max rank Genius soldier on a mission to raise their intelligence to Savant you'll get an extra 21 points out of it.
If you're looking to get more AP out of covert ops, it's best to only use "raise intelligence" for soldiers that are already Gifted or Genius, since the bonuses increase more. Otherwise it's better to just do the "ability points" missions to get more points into the shared pool.
For your last question, I believe there are several mods you can find that show the whole skill tree. I think there might even be just a config edit you can do, but I don't know enough to tell you for sure.