The whole point of a hypercomputer is not being a Turing Machine. So yes, it can calculate things that are TM-uncomputable.
You can't use it to build a self-contradicting Turing Machine, because a hypercomputer cannot be emulated by a Turing Machine.
There is no "the" halting problem. Each class of machines has its own halting problem. The halting problem for turing machines is very hard. The halting problem for non-cyclic finite state automatons is very easy. A hypercomputer that can solve all halting problems is self-contradictory. A hypercomputer that can solve turing-or-weaker halting problems is not self-contradictory.
You can't use it to build a self-contradicting Turing Machine, because a hypercomputer cannot be emulated by a Turing Machine.
There is no "the" halting problem. Each class of machines has its own halting problem. The halting problem for turing machines is very hard. The halting problem for non-cyclic finite state automatons is very easy. A hypercomputer that can solve all halting problems is self-contradictory. A hypercomputer that can solve turing-or-weaker halting problems is not self-contradictory.