Proving properties and computing values are quite different things, and proofs can absolutely be done on Turing machines, e.g. with proof assistants like Lean.
no, see the problem is that the machine needs a well defined problem, and the "BB(n) grows faster than any defined problem" is well defined but you would not come up with an insight like that by executing the BB(n) function. that insight requires a leap out of the problem into a new area and then sure after it is defined as a new problem you enter again in the computability realm in a different dimension. But if the machine tries to come up with insight like that by executing the BB(n) function it will get stuck in infinite loops.
