Which has a greater influence on sound quality, the amp or the preamp?

Yes, but there is always a point when it is time to upgrade speakers too.

Yes, I am sure there is.

Yes

Which is more important to walking? The left or the right leg? 

Now that I've gotten my morning snark out of the way, my answer is that they are both critical but they perform different functions and thus their relative importance is contextually dependent. 

I think it depends on the equipment used so Both.