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.

I am pretty sure this identical question has been asked in the past, and every time I have read it, the answers are (like almost all audio topics here) split with no clear consensus, and a lot of "both" answers.

I am going with "both," but the importance of each manifests itself in different ways. I’ve played around with a few different tube amps, and each amp change did something. I didn’t start to feel like my system was sounding truly "musical" until I upgraded from a digital HT preamp to my first tubed preamp. I upgraded that to a better tubed preamp and lost some warmth but gained air and detail (which to me means a better imaging and a better defined sound stage).

 but you would never know how good your speakers would/could sound if you don't give it a chance to be supplied with good input.

+1

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

Not a good analogy.