Would hypercomputation machines be capable of simulating/computing/programming everything?
I think you have an important misunderstanding about what "uncomputable" means. It doesn't mean "doing something mathematically impossible", such as producing a non-trivial factorization of a prime number. There simply are no integers other than $1$ and&nbsp;$181$ that divide&nbsp;$181$ and it doesn't matter what method of computation you use, you will never find these non-existent factors because they just don't exist.  The factoring problem is not uncomputable: you can write a computer program that will correctly factorize any number that you give it. In the case of&nbsp;$181$, the program will just say, "The factors are $1$ and&nbsp;$181$ and that's all."

A problem is uncomputable if there is no Turing machine ("computer program") that can solve it. The standard example is what's called the halting problem: you have a computer program and you want to know if it will terminate, or get stuck in an infinite loop. Turing's famous result is that there is no Turing machine that will take the description of another Turing machine as input and correctly say either "this terminates" or "this loops forever".

Some models of hypercomputation can solve this problem. That is, the "hypercomputer" can decide whether given Turing machine will terminate. However, Turing's proof is very general and applies to any sufficiently powerful model of computation. So, although the hypercomputation model might be able to solve the halting problem for Turing machines, Turing's proof can be easily adapted to show that the hypercomputer cannot solve its own halting problem. That is, there is no hypercomputer program which takes as input the source code of another hypercomputer program and correctly determines whether that program will halt.

In summary, every sufficiently powerful model of computation has uncomputable problems, so no model of hypercomputation can compute everything.