Following the comments you've received, I'll add my own subjective answer in the affirmative to your first question. I think we've already compiled enough votes and comments here that support @JoshGitlin's unscientific answer to say that there is some empirical basis for theorizing the existence of a "taste" acquisition process in music. Nonetheless, here's another somewhat unscientific (and inadvertent) contribution from Wikipedia:
Acquiring the Taste was the second album of English progressive rock band Gentle Giant...This was a departure from the blues and soul styles found on their self titled debut. It was more experimental, more discordant, and with more varied instrumentation. In the sleeve text, the band made this famous declaration:
"...It is our goal to expand the frontiers of contemporary popular music at the risk of being very unpopular. We have recorded each composition with the one thought - that it should be unique, adventurous and fascinating. It has taken every shred of our combined musical and technical knowledge to achieve this. From the outset we have abandoned all preconceived thoughts of blatant commercialism. Instead we hope to give you something far more substantial and fulfilling. All you need to do is sit back, and acquire the taste."
So there you have it: a fairly sophisticated band of musicians appealing to some common intuition about the very process in question, acquiring a taste for certain music. At the risk of being flagged for spamming, I'd even recommend that anyone curious enough to try out the process for oneself should give the album a listen; it's likely to be quite different from the music you're used to, but it's also likely to grow on you somewhat (it's certainly grown on me since I first heard it)!
As for your second question, generally speaking, aesthetic appreciation of a given, enjoyable stimulus (including, but not limited to discrete musical compositions) changes in at least two ways over repeated exposures.
- Hedonic adaptation often occurs to some extent, such that as novelty wears off, so does enjoyment.
- Nostalgia plays a somewhat opposite role under certain circumstances, such as those that associate the stimulus with a desirable memory.
There's probably plenty more to it than just these two influences though. How it all actually plays out would be very interesting to see, so here's an upvote for any references others might be able to offer in their own answers!