You can’t really ‘experience’ mathematical proofs in any meaningful sense. You can understand them, but I don’t think you could describe that as any kind of experience. ‘Someone asked me what 8 plus 7 is, and I experienced the answer as 15’. I don’t think so.
They’re all very big questions deserving of an essay-length response. My view, which is not mainstream, is that the reality of any object in a general sense does always include a subject for whom the object of definition is real. That is Schopenhauer’s ‘no object without a subject’. But it has to be interpreted carefully. It shouldn’t be taken to mean that ‘the thing ceases to exist when not perceived’. When it’s not perceived or considered then plainly nothing can be said about it but that doesn’t imply its non-existence. Imagining it not existing is a projection. (I’ve written an essay on this topic, The Mind-Created World.)
Metaphysical realism, as it is called. This is the aspect of quantum physics that Einstein could never accept. Wheeler said, in another paper Law without Law, that
The dependence of of what is observed upon the choice of experimental arrangement made Einstein unhappy. It conflicts with the view that the universe exists “out there” independent of all acts of observation.
No elementary phenomenon is a phenomenon until it is a registered (observed) phenomenon. …A phenomenon is not yet a phenomenon until it has been brought to a close by an irreversible act of amplification such as the blackening of a silver bromide emulsion or the triggering of a photodetector. In broader terms, we find that nature at the quantum level is not a machine that goes on its inexorable way. Instead what answer we get depends upon the question we put, the experiment we arrange, the registering device we choose. We are inescapably involved in bringing about that which appears to happen.
The lesson of quantum mechanics is that before that ‘irreversible act’, it can’t be said what, actually, exists in terms other than the probabilistic.