I don't think the number of transports affect invasion times anymore. Last I checked, invading colonies takes 2 turns, and core worlds take 5. I think resistance planet improvements affect invasion times by increasing the number of turns it takes. Any combat ship can take a colony, but you need transports to take core worlds.

You are correct.  Throwing more meat at the problem reduces the time it takes to conquer the planet.   There is a minimum amount of time, and using soldiers with higher resolve and a higher soldiering skill will also reduce the time.   Don't worry though -- if you are Mimot, you can just make more, quickly.