1) I just built the Spire Galactic capitol. I was under the impression my ultimate goal was to unlock an AI controlled Spire that would just attack and wipe out the enemy AI. Am I wrong on this? Do I control the Spire the entire time?
You're not actually done building up. Once you have the capitol it will scan for the last shard with which you build the thing that leads to that last stage. You also have the option of using the new tools from the capitol (being able to build as many spire colony ships as you want, BB's, DN's, advanced modules, and just take over the map despite all the AI can throw at you).
2) The Spire City hub can support three structures (reactor, habitat and shipyard), however, it says that when the hub's development level increases it can support more. How do I increase this development level?
One of the journals is supposed to tell you, and your first city has to reach the second level (and thus be able to support 6 structures) to scan for the second city shard.
Anyway, you have to have a hab center at the city and clear all neighboring planets of AI command stations, and the hub will be able to support 6 instead of 3.
3) This is my 5th game(lost previous 4), I'm playing two random 6 AI's, I'm 13 1/2 hours into the game, own 16 planets and the AI progress is 740. Is that way too high? I still need 18,000 research to unlock the Spire dreadnought and battleship......
740 is a tad high, and you're maybe an hour or so later at getting the capitol than I was targeting, but if you're playing on diff 6 the normal waves are roughly half the size they are at diff 7 and the event-attacks from the spire stuff are only 90% as big as the diff 7 ones. So if you're holding now you should be ok. If you need knowledge, just start glassing AI planets as necessary