1) Can AI recapture a planet?
In other words; does AI use colony ships or not?
No, the AI will never recapture planets it may have lost, no worries there.
2) At the beginning of a match you can see what special ship you can get. Do all of the other possible homeplanets/special ships also represent the ones you can caputure later in the game (if only choosing 1 homeplanet) or is it random after you pick the first?
The new ships types you get while capturing Advanced Research Stations are totally random, not related to the pool of ships you can choose in the starting planets.
3) Lastly, how exactly does AI resource management work?
- Does AI share between eachother?
- Is resource income global or spend more locally?
- Do they spend most of what they get or does AI at the end have near infinite resources (Because they reach the ship cap)?
AI doesn't use resources, they get reinforcements on their planets at a regular interval, the amount depends of the AI Progress, the difficulty level, and one special building (don't remember the name of it, but it's pretty rare). The AI will then dispatch these reinforcements on the most important places, most usually on the planets next to your territory. You can see witch planets are more likely to get reinforcements by checking their "alert status". If they are on alert, they are more likely to be reinforced.
For attacking, the AI get "free waves" at some points, can use it's garrisoned units to attack you (during a cross-planet attack or if you anger it by some way like destroying a Command Station or Raising an alarm when an Alarm post is near - Prerelease only) or you can be attacked by spec ops who roam the galaxy and sometime attacks you without warning. The spec ops aren't too dangerous however. Make sure to defend all your borders.
I hope that helped !