Why it is so hard to make a nondeterministic computer? If such a device is physically realizable then would that be a counterexample to the extended Church-Turing thesis?
Is it possible to make a machine that when it encounters a non-deterministic step it takes an arbitrary one? Yes easily.
Will this machine be able to solve NP problems in P? No because it will most likely pick the wrong path through the execution.
Similarly for outputting a number there is no way to force the machine to match the required path needed to output said number.
Do you mean "indeterministic" in the sense of quantum mechanics? It's easy to build that kind of indeterminism into a computer, by incorporating a routine that goes and checks the state of a quantum mechanical physical source. That doesn't necessarily violate the Church-Turing thesis, since the source of quantum mechanical randomness might not be the result of an algorithmically realizable process. In that case, if it turned out that what the quantum mechanical source produced was uncomputable, this wouldn't violate a Church-Turing thesis that says that any algorithmic process can be computed by a Turing machine.
There are discussions of a variety of versions of and extensions of the original Church-Turing thesis, and discussions about whether natural processes or specially constructed experiments are ever counterexamples to these versions. Jack Copeland has useful papers on this. Not everyone agrees with him about everything--I don't think I do--but I think that papers such as his "The Church-Turing Thesis: Logical Limit or Breachable Barrier" provide good entry-points into this literature.