it should be possible to stop improvements from auto-upgrading
It is, but you have to do it planet-by-planet. The option is on the govern planet screen.
I could be looking in the wrong places but Economic Starbases don't seem to affect either Research or Production outputs, only Wealth.
You must not have researched techs that grant the modules yet because they are in the game. Also, base economy starbase gives +10% production which adds to research, wealth, and manufacturing.
Buying a ship should produce it instantly instead of next turn.
In order to help balance multiplayer buys of ships and improvements occur the turn after you pay for them. Unfortunately, I don't think they'll be going back on this.