Thanks for the fun rabbit hole. They can't really solve the halting problem though, you can make an oracle solve the halting problem for a turning machine but not for itself. Then of course you can make another oracle machine that solves the halting problem for that oracle machine, and so on and so forth, but an oracle machine can never solve its own halting problem.
They exist in the same grammatical hierarchy so theoretically they can solve the same problems. What I should have said was that nondeterministic turing machines can solve NP problems in P