What interesting open mathematical problems could be solved if we could perform a "supertask" and what couldn't?

The infinitary computability concept that you describe sounds fairly close to the concept of infinite time Turing machines, which extend the operation of ordinary Turing machines into transfinite ordinal time, and thereby provide a robust mathematical model of supertask computation. The basic idea of the ITTM model is to allow an ordinary Turing machine to operate into transfinite ordinal time, by defining the limit configuration and allowing it continue its operation. At successor ordinals, the machine operates according to the rigid instructions of its finite program. At a limit stage, the head is reset at the left-most cell, the new state is a special limit state, and each cell of the tape is updated by taking the $\limsup$ of the values displayed in that cell going into the limit.

The principal reference is my paper with Andy Lewis: J. D. Hamkins, A. Lewis, "Infinite time Turing machines'', J. Symbolic Logic 65, 2000, p. 567-604. The machines were first defined by Jeff Kidder and myself.

In the ITTM model, we investigated both the complexity of the sets that are infinite time decidable, as well as the lengths of time that such computations take. It turns out that not only are all arithmetic sets infinite time decidable---and so all the arithmetic questions of number theory are decidable by such machines---but the collection of infinite time decidable sets reaches nontrivially into the projective hierarchy, beyond the hyperarithmetic sets $\Delta^1_1$ and indeed beyond the lightface analytic sets $\Sigma^1_1$ and into the $\Delta^1_2$ sets. So much more complicated questions can be settled by these machines. For example, with infinite time Turing machines one can test a countable order as to whether it is well-founded, which is in general a complete $\Pi^1_1$-test. In the ITTM context, there is a very interesting theory concerning the supremum of the lengths of the halting computations in comparison with the lengths of the stabilizing computations in comparison with the lengths of the not-yet-repeating computations. The ordinals where these phenomena occur are known as $\lambda$, $\zeta$ and $\Sigma$, and they are intimately connected with certain fine-structural features of the constructible hierarchy. For example, Philip Welch proved that $L_\lambda\prec_{\Sigma_1}L_\zeta\prec_{\Sigma_2}L_\Sigma$, and the ordinals are characterized as the least triple of ordinals with that property, a result that has become known as the $\lambda$-$\zeta$-$\Sigma$ theorem.

The arithmetic sets are exactly those sets that are decidable uniformly in time $\omega^2$, with each limit giving you essentially two additional arithmetic quantifiers. The hyperarithemtic sets are exactly those sets uniformly decidable in time less than $\omega_1^{ck}$, the supremum of the computable ordinals. Meanwhile, the clockable and writable ordinals reach far higher than this.

Not all sets are infinite time decidable, of course, since there are still only countably many programs and thus only countably many decidable sets. Indeed, there is an infinite time analogue of the halting problem, which is infinite time semi-decidable but not infinite time decidable. There is also an analogue of the Turing degrees for this context, with two jump operators, corresponding to the boldface and lightface halting problems, for which one either allows or does not allow real parameters.

You may consult this list of my further articles on the topic, whose bibliographies contain references to the increasingly large literature on the topic. There are now infinite time analogues of computability theory, complexity theory, equivalence relation theory and much more.

You would be able to check whether our axiomatic systems are consistent or not, since a contradiction is a deduction of finite length.

Forever Is a Day: Supertasks in Pitowsky and Malament-Hogarth Spacetimes discusses about hypercomputation of Fermat's Last Theorem -albeit certain technical background is needed to understand the details.

You can also google Malament-Hogarth spacetime. Here is another paper that discusses the question.