Ships repair slightly faster at shipyards and slightly faster still at starbases. There's also icon to the right of the ship/fleet window in the lower left (when the ship/fleet is selected) that you can click on to speed up repair also, that one puts the ship/fleet to sleep until you cancel it or it is fully repaired.

As far as I know you can still upgrade the Orbital Academy to, for example, the Top Gun Academy, and then build another Orbital Academy on another core world. But there are a limited number of upgrades, so you can't build them on colonies anymore and you can only build a few on your core worlds. Plus the hit point bonus applies only to ships built on that core world, negating the exploit that got players (and AIs) to hundreds or thousands of hitpoints.