Are all undecidable/uncomputable problems reducible to the Halting problem? Theory of computation tells us that there are some languages that cannot be recognized by a Turing machine. That is, there are well-defined problems for which no Turing machines can provide an algorithm which solves the problem.

Many problems that are undecidable seem to have a reduction to the Halting problem: Matyasevich's proof of the undecidability of Diophantine equations uses the Halting problem, the machine equivalence problem uses the Halting problem as well...

The question is: are all undecidable problems (in the sense of Turing) reducible to the Halting problem?

If so, is there a proof of this? If not, is there a counterexample of such a problem that provably cannot be reduced or mapped to the Halting problem.
No. As I recall a practical example can be made out of a function inequivalence statement. More abstractly, there is something called Turing-degree in undecidability and infinite levels of "hardness".  Each level has it's own Halting Problem strictly harder than those at every level lower.  Check out the first section of this Scott Aaronson lecture and what he calls a "Super Halting Problem".