"The network needs machines that store both application data and user data, and rewards machines that perform this storage with cryptocurrency tokens." - Isn't it likely a bunch of people will just host on AWS/Google Cloud?

Yep, I think that is likely. In fact it is encouraged. The codebase has instructions for running on Google Cloud via Kubernetes:

https://github.com/axiom-org/axiom/tree/master/testnet

Being decentralized means that the same backend logic can run on any cloud provider, or on a simple server connecting to the internet. But it's okay if in practice many of the hosting providers are running on top of cloud providers.

Why use Stellar for consensus? Why not something like Tendermint via the Cosmos SDK? Wouldn't you want your token to be used for node membership through a PoS model? I just don't see what advantages federated byzantine agreement has over BFT through PoS.

How does the network make sure that a hosting provider that claims to be hosting is actually doing so?

It's up to the client. If a host isn't providing good enough service, the client can switch hosting providers.

Mr. Lacker, congratulations on this, it looks wonderful, a kind of practical libertarian Banksy.

An obvious use would be microblogging platforms. What structural obstacles remain for such an app on Axiom to become competitive with Twitter, etc.?

The biggest obstacle for any new social media application is the network effect. There needs to be a reason for a small set of people to use it in the first place. That's kind of app-specific, though, and it's a product-centric issue. I think with Axiom the backend technology is ready for such decentralized apps to be built.