Yes. To conceive of a museum of fine arts, you also need to have a culture that has a sense of different relative worths of both artistic styles and art artifacts. What belongs to a museum vs what doesn't, what is fine art vs what is commercial crap or kitch, and so on. One that believes in informed curation - curation being the very core of what a museum does.

Of course this requires an era where the opinion of people devoted to studying, discussing, and creating fine art isn't supposed to be equally respected to that of Joe Sixpack, Jaqueline Middlebrow, Joy Belieber, Arthur Incel, or Random J. Person, where everything is up to personal taste.

Yes, by their own assessment following the changing or expansion of their tastes.

Why change, grow, or learn if you don't think the state after is better than before?

I don't care for the use of terms that imply appreciating the fine arts is somehow on a "higher plane" and places those who do so above those who don't, but I do think it's worth acknowledging that it takes significant time and effort to develop such appreciation, and that the rewards are subsequently substantial. Personally I've invested significantly in the study and appreciation of Western classical music and I believe what I've gained from that could not be achieved simply by listening to top 40 hits on the radio - but I don't believe those who do only the latter are in anyway "lower" on any sort of ranking system, just that their brains are wired differently, and their cultural background etc. is such that it probably wouldn't make sense for them to make the same investment into music appreciation that I have. It's almost certainly true that they've invested considerably into appreciating other worthwhile aspects of life that I haven't - we can't do it all!

Yes, it's called aesthetics.