I'll just say this - at least some of the ideas you're interested in have been approached by homotopy theorists, like Jacob Lurie, and by category theorists. Unfortunately, this seems to be primarily done in the language of infinity categories/infinity topoi (of which the category of chain complexes of sheaves of Abelian groups on some site is an example). However, I'm pretty certain there must be an affirmative answer to your question. One place to start might be ncatlab.org/nlab/show/infinity-gerbe.

Hm... maybe you did already.

Could you say what it means for a stack to be a gerbe "under a sheaf"? I'm not familiar with that terminology.

The short answer, for the record, is no. Nobody has done this.

So, random, possibly dumb question: Can I think of Steenrod operations like some kind of lambda ring structure?

Furthermore, that person is Tyler Lawson, and he pointed me towards Brooke Shipley's paper on Morita theory: homepages.math.uic.edu/~bshipley/tilting7.pdf where you want to use the fact that a descent datum is the same thing as an $S\langle G\rangle$-module.

For posterity, I should add that someone just told me that faithful Galois extensions of ring spectra are of effective descent for modules, though I've not yet found a reference.

Thanks anonymous. I thought that might be the case, otherwise I don't know why he would call it that. Presumably a weaker condition holds as well.

I won't be there, unfortunately.

I think Urs' answer is interesting and a good one, but I also want to add that in some sense these really are the same thing in the sense that to any topological space one can associate a Grothendieck topology. Where the difference lies, in my mind, is in the notion of having "enough points." There is quite a bit of subtlety here though that I'm not an expert in. I recommend looking at locales however. A good place to start might be ncatlab.org/nlab/show/locale.

You should swing by the homotopy theory chat room sometime. Those guys in there chat a lot (and know a lot) about TMF. It's not really my specialty. chat.stackexchange.com/rooms/9417/homotopy-theory

I'm not sure about solving classical topological problems. The reason that I consider it to be interesting is because it's supposed to be to the second chromatic level (of chromatic homotopy theory) what K-theory is to the first chromatic level. The reasonn, in my mind, that this is interesting, is because the chromatic picture in stable homotopy theory happens to look a lot like the derived category of a Noetherian ring. That's sort of vague, but I think it speaks to some of the deepest structure of algebraic mathematics.