I've been wondering about this for a while too. In the direction of a topological Jacquet–Langlands correspondence, recently we have this paper. You will also find some pitfalls in this theory. For example, if you believe that the Lubin–Tate tower should in some sense incarnate a local Langlands correspondence, the fact it does not naïvely descent to the sphere is a problem. To my knowledge, Andrew Salch (the auther of the paper I linked) is the person who thought about this type of topological Langlands correspondence the most, I can recommend also reading his other papers.
You mention you recently started learning about homotopy theory, so maybe this recommendation is too advanced, but yes, there is a theory of spectral abelian varieties and also a theory of moduli of spectral elliptic curves. For this you should read Lurie's series of papers on elliptic cohomology.
There is also a notion of topological automorphic forms, however to my knowledge it is not clear, what should happen on the Galois side. Classically one would consider representations of the Galois group of $\mathbb{Q}$ (to be more precise this only sees cohomological automorphic forms, but to my knowledge topological automorphic forms so far also only generalize cohomological automorphic forms), the rationalization of the sphere is just $\mathbb{Q}$ however (and in any case the étale site is nil-invariant and cannot see the difference between the sphere and $\mathbb{Z}$, so one has to do something smarter than étale cohomology). Even with recent advances in categorical Langlands theory none of the proposed candidates on the Galois side that I know of seem to make sense or yield something new when considered over the sphere.
Also note that spectral algebraic geometry is a very different beast from derived algebraic geometry, and so far it seems to me that advances in elliptic cohomology yield applications to chromatic homotopy theory. So far I have not seen an application to number theory.
