You currently keep ship upgrades until the next wipe. This includes if your ship is stolen, you crash it, or it gets destroyed for any other reason. Next time you spawn the ship, it'll have all the components on it at the time it was lost.
On the ship upgrades, the power plant produces a ton of heat, so you should improve the coolers first. But, so long as you're power plant can run your current ship and components, there's currently no benefit from having a larger power plant. You're probably better off not upgrading it.