Only the tech the bonus is attached to will grant the bonus. Buying an advanced colonization tech will get you the 50% usage it normally grants, but you still need to research or buy the basic tech to get full use of those planets.
Usually the earlier tech is worth researching only if it has a passive bonus or another tech it leads to (ex you steal beam weapon theory without having space weapons, you would need to research space weapons to get the other weapon trees). Of course, this can occasionally lead to hilarious AI exploits - gift an AI Impulse II engines and they generally won't go back to research Impulse I to get the passive bonus. That may have been changed since TA 2.0, I haven't tried it recently.
AFAIK the only requirements are 1) you invade, not spore or flip; 2) the tech is stealable (many are not); 3) they have the tech, and you do not. However, don't expect to steal a tech each time you invade, even if you meet all the requirements to do so. There is a random chance to do so each time you invade. I've never really tested it, but it seems your odds to steal go up the farther behind you are in tech, meaning stealing one tech out of 20 possible is more likely than stealing the only tech available.
