Carrier only fleet? No, a massive fleet of flak will stop it though you would get some ships destroyed out of it.

Best? Well Advent carry the most per supply (6.7 per strike group vs 7 per with TEC and Vasari) but after that nothing noticable. Vasari have the best overall fighters/bombers (high hitpoints and armor with powerful late game weapon phase missiles).

Research? Weapons first. The initial strike upon a high profile target (high level cap or starbase) can change a battle quickly. After that strike craft can be killed quickly with or with out the armor/hitpoint research.

Carriers have numerous advantages and disadvantages.  The biggest problem they face is that they're just too easy to run down and kill with frigates.  Light frigates in particular are fast enough to close the gap before you can blink, and early-game this is going to put a quick end to your carrier-based strategies.  In the late-game, huge swarms of strike craft get countered by the right capital ship abilities.  The Vasari in particular can completely shut you down with jam weapons and phasic traps.  Advent and TEC will punish you with TK push and flak burst respectively.  Going pure carrier is just asking to get countered.  If you can substantially outnumber the enemy this is somewhat irrelevant, and carriers (equipped with bombers) are definitely the game's best coup-de-grace unit.  However, for a straight fleet battle I'd be inclined to mix unit types thoroughly and wouldn't want to lean too heavily on strike craft.

As for the best carrier, they're all roughly equal.  The Advent carrier has better range due to its larger antimatter reserves (this scales nicely with the larger unit size) but otherwise they all cost and perform about the same.  Now, as for strike craft, the biggest difference is Vasari, which uses the phase missile type weapons for both its fighters and bombers.  This makes them incredibly effective against anything that has shields, and absolute monsters against capital ships.  When fully upgraded, Vasari fighters are better against capital ships than the other faction's bombers. 

If you really push it late game and can max out mass transcendance, the Advent Halcyon becomes an extremely nice carrier.  It can swat away enemy strike craft with its TK push, and offers six squads thanks to its adept drone anima.  Level it up once, and now it's fielding eight, outpacing an equal value of carrier cruisers, and level it up twice and anima tempest will really make your day.  Of course, I've almost never seen it done in multiplayer.

Matt that is a nice fleet but a costly fleet.

Also 1 level 3 concentration aura is better than 3 level 2 concentration aura you’re better off getting a Halcyon and/or a progenitor then more than one capital ships that boost strike craft.

The problem with domination is that you need to be in combat to use it, meaning you'll be taking casualties anyways as the fight goes on.  If you're actually increasing in fleet size from using domination, that means you're not taking casualties in combat, and you've pretty much already won in this case.

Domination is highly overrated.  It's worth it to grab hoshikos and overseers, since Advent lacks any decent form of hull repair, and in the mirror match it's good against guardians since you can then let off a repulse right in the middle of the enemy's fleet, but that's about it.  Most of the time, you're better off conserving antimatter for vertigo which is a downright awesome ability once maxed out. 

I'd be more inclined to have only 1 or 2 Raptures tops, and spend most of my capital ships on Progenitors and Halcyons.  A couple Radiances and Revelations aren't bad, but use them sparingly since they're only good for knocking out enemy capital ships, which is something a sufficient amount of frigate-based firepower will do just as well.