How much does quality affect the animation speed? Does it speed up the door animation too?

I made some rare silos but I didn't notice any change in game.

The rocket raising speed scales with the rocket silo crafting speed, so rare should be 160%; about two seconds faster. You won't easily notice it if you're not comparing it to a normal rocket silo.

Since rockets aren't affected by speed modules, its much less pronounced a change compared to the crafting speed. There is a way to fix that in theory, but it is infeasible to have one rocket entity for every potential speed combination, especially with the beacon changes. The engine has a hard limit on how many different entities can exists, and it is WAY less than the 145000 entities doing it this way would require (if one wanted 100% accurate results).

The two-and-a-half times increase on the legendary rocket silo is much more pronounced, and saves you about eight seconds compared to the common rocket silo though.

Is it 2 seconds off the entire launch animation or just the rocket raising into position? If it's the whole launch animation that would explain why it's hard to see at a glance.

I measured that time from completing the rocket to it finishing raising. The launch should be similarly sped up but I didn't measure it directly, but the appropriate values are set.