These answers are not super-in-depth, but I have tried to at least answer them.
1. Have you ever played a KDF captain with the Vo'Quv? If so, you may already have an idea of the kind of builds that mesh well with this carrier. That said, briefly -- as I feel there are others who have explored in more depth where spending 6 points is better than 9, etc. -- take note of the science boff slots and the ability to emphasize them. With the Atrox layout, skills like TSS, HE, GW, Tyken's Rift, Science Team, Eng. Team, EPtS, and so forth will be strong, so keep that in mind when spending points. As an engineer, your build for the cruiser you have may still be applicable in many ways, though if you haven't spent much in the science areas it is probably worth re-considering it.
2. Surivability is still great for the carrier, but yes definitely explore science officers. Look into Transfer Shield Strength, Hazard Emitters, Tractor Beams and/or Tractor Beam Repulsors, Gravity Well, Tyken's Rift, and Science Team. For PvE, anyway. You may want Polarize Hull if for some reason you think your carrier will be getting close enough to cubes/gates or the occasional sphere to warrant wanting out of tractors.
3. Think of it as two cooldowns -- the one for the skill you've just used having to recharge and the one for all skills that share a system. The latter is typically short enough that you can rotate two of the same skill. A common example is Emergency Power to Shields. By the time the first one fades, you can cast the second one to keep it up indefinitely. In fact, you can usually keep two emergency powers up fairly constantly (a good example may be a cruiser keeping EPtS and EPtW up) by keeping on top of recharges/cooldowns. TSS and HE also can be carried in pairs to better assist teammates/yourself. Some doffs can cut the recharge further.
4. No. They don't have a limited number of charges or anything like that. The hangars you equip are like consoles/weapons in that they won't need replaced (unless you are upgrading or swapping which ones you are using).
5. Typically the advanced fighters serve this role well, but the Atrox has a lot of options. Consider the Danube runabouts for their tractor beams to annoy/hold targets or other runabouts for their increased hull over fighters.
There are some starting points here. Generally craft with torpedoes and some tactical skills will "strike" harder than those with just some beams or a special ability, but sometimes the debuffs are totally worth it! Consider mixing as well and having two different hangars set up.
6. Beams and turrets, yeah. However for PvE the Borg are nice enough to fly in predictable paths or without too much evasion to make a forward arc viable (dbb, cannons, torps). You may be doing this anyway to fire off gravity well or the like (though you can easily turn the ship when needed otherwise). If you stick with beams and consider the tetryons like it comes with (though upgrade these, of course, as they are terribly low quality), look into the Power Siphon skill to help with the shield drain. This may also boost polarons.
7. Go to the Academy on earth and talk to the various NPCs to recruit more. You can also trade, like, 5 of one quality to get 1 of a better quality if you want. Lastly you can purchase packs in the Cstore, but that's dumb.
Also, as you travel to different sectors, different missions will be available. And after an amount of time the ones in the current sector will update/change.
8. Some of it is. I realize I am a f2p player and wasn't playing under the previous system, but long story short you'll want to take a look at the STF rewards. Roxxy has the Borg gear (under "general" when you talk to her for merchandise) set pieces, though you'll want to make sure to run the mission "Assimilated" in the Undine/Borg missions to get the console (do this on every ship; it's a great console). You can also get the MACO and Omega pieces as a Starfleet officer. You will likely want to set up with the MACO shield at least and then fill the rest of the slots with Borg pieces for the carrier. Don't ignore the potential of the MACO deflector, though.
You can also get MX X, XI, and XII weapons through STF drops, though all of these carry a [Borg] modifier that you may not want for general PvE/PvP.
9. Yes
10. There's a dilithium shop that you can access from the dilithium button under the mini-map in the UI. It's a great way to get ship consoles if you don't want to spend the EC on the exchange / have dilithium to spend. It's also where you get the advanced hangars, which are not tradeable among players. So save your dilithium for your fighters and shuttles, etc. first.
The STF rewards are the best/easiest place for shields/engines/deflectors and weapons for PvE.
11. For my eng. character, I so far have played around with 6 beams, Neutronium Alloy and an EPS console for engineering (though I dump the EPS for more armor often, as I don't need to boost for swapping energy levels too much on the engineer) | a Field Generator or two, shield emitter array, and the Borg console for science | two appropriate energy consoles for the beams I am running.
A more healing-focused build may dump weapon power/boost to use more universal consoles (i.e. ones from Cstore ships) instead of tac consoles, but *shrug*.
12. As I understand it, open the Cstore and look under the unlocks section (last section in its menu). I believe veteran rewards are there just like unlocks from some of the retail packages.
13. I would run STFs to get the rewards so you can equip the ship right away. However, I am pretty sure STFs don't net you XP. Look out for the Mirror event, as that's a nice XP boost, and replay any mission that either you liked or that has a reward you might want (as rewards scale to your level).
14. Redshirt is for Season 6 testing.
15. I suspect there may be plenty of fleets for that here in STO and wish you well. I myself cannot speak from experience, as most of my characters are in a joke fleet made with some friends. I PuG all the time and have yet to find a "home".