So I have heard here and there that the PLAN has sub bases inside of sea caves on Hainan. Is this true, or should I say is there any good evidence to support this?
Plenty. A lot of the pictures you see on this thread are in Hainan. There is a large sub cave entrance there that you can see in GE. I won't say, that a base resides inside a cave, but a few docks for sure. The real base is what is around the cave and around the bay. It looks much more like a resort than something from a James Bond movie.
From what I have read, the LA class is not noted for being a major step forward in sound reduction over the Sturgeon/Thresher. Rather, its just a bigger, faster, more capable sub.
Bigger allows for better sound reduction. Speed is life underwater.