I deleted my last comment because I convinced myself it was wrong, but now I am convinced it is ok again. I think you need an isomorphism of $E$ and $F$ on $X\times \{1\}$, and not trivializations. Conner and Floyd use this definition of $K^0$ (without the virtual dim=0, you might need to remove that, or maybe these cancel each other?) in their book The Relation of Cobordism to K-theory

probably Lunts and Orlovs paper has something about this. I'll check and get back to you

DG means differential graded. This means it's enriched over chain complexes. By the Dold-Kan correspondence this will give you an infinity category. As for why infinity categories help here, the idea is they make homotopy cofibers, and homotopy colomits in general, functorial. This is because they let you treat commuting up to homotopy as commuting, and let you treat rather complicated classes of weak equivalence like they were homotopy equivalences.

As for why someone with a homological background might want to know about higher categories, there is this very annoying fact that the cone of a map is not a functorial construction. DG and infinity categories fix this

This might also give a connection to the Blumberg-Cohen-Schlichtkrull result, since Atiyah duality gives that $\Sigma_+^\infty M$ and $Th(NM)$ are dual.

If $X$ is finite CW, then it is dualizable in spectra, so that $F(X, E)\cong E\tensor DX$, and then $THH(F(X,E))\cong THH(E)\tensor B^{cyc}(DX)$, where $D(X)\tensor D(X)\to D(X)$ is the dual of the diagonal. I don't know what is known about the $B^{cyc}(DX)$, but it seems approachable at least.

@OlivierBenoist topological cyclic homology

not that I know of, but I am far from an expert. Sorry.

The Ax-Grothendieck theorem gives that when $p=q$, then injectivity implies surjectivity.

@DenisNardin I understand why both operations separately are $A^\infty$, I just don't understand the distributivity I suppose.