Oh gods, yes, please let me disable the damn FATE music. I don't mind FATE farming for the shared FATEs in general because it gives me more reason to hang out in the beautiful areas of this game. However, if this wonderful atmospheric music is constantly interrupted by DAAAA DAA DA DA DAAA DA DAAAAAAA whenever I join a new FATE, it's really annoying and usually ends up with me turning the background music off entirely and listen to something own or Youtube or start a show on Netflix. I already turned off regular battle sounds for the same reason, please let me do the same for FATEs. (Also gives me Eureka flashbacks...)
I don't do Levequests, so I can't tell a lot about that, but more options is always good, and by what you say it's just as annoying, so +1 for that as well!
Not really sure if a lot of people are going to see this post though, considering there is an area for FATE feedback:
https://forum.square-enix.com/ffxiv/forums/650-FATE
Maybe consider moving it there, if this is possible, or open another thread over there and edit a link into your original post?