Sorry, I misspoke. The theory of gravity is actually a really interesting physics problem, but the fact that gravity acts on objects is pretty much undisputed.
Hardly! If you put an object in empty space, without any other objects nearby, it will be completely unaffected by gravity. Unless the object is large enough that its own gravity can overcome its structural or material resistance to being crushed. Further, unless an object is actually in contact with another object, it is unaffected by gravity from its own reference frame. The apple falling out of the tree (neglecting air resistance) has no way of knowing that it is falling.
I’m sorry, I’m not sure what to make of this: whether or not GR (our current ToG) is correct, and, if not, what’s missing or how it might be wrong is a hugely important active field of research.
As is, of course, trying to integrate QM and GR.
They aren’t ossified, they are hard. Conceptually hard. Mathematically hard. And a few other hards.