Can hypercomputation compute all kinds of incomputable numbers/functions/problems…etc?
It is a little hard to say for sure since since one has not found a final Theory Of Everything in physics. Mostly it seems that hypercomputation is physically impossible.

On the other hand, an amusing development is the work of Nemeti et al. on using time travel (closed timelike curves) to do hypercomputation.

The so-called Malament-Hogarth spacetimes are known for exactly this --- that one can compute forever in them and thus solve the halting problem.

Hogarth worked on this in the 90s and Welch showed that even a larger class of problems than those solvable using the halting problem (all hyperarithmetic problems) can be solved in MH-spacetimes. This does not include all oracles though, as there are uncountably many oracles and only countably many hyperarithmetic ones.

(As for things that are logically impossible, like Russell's set, they are also physically impossible since "physics obeys logic".)