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 naively 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 topological Langlands correspondence the most, I can recommend also reading this 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 me 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 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.