Correct. You'd send the DMCA to Apple to force Apple to block access to the app, no matter where the developer resides.

No, and that's where/why the whole "service provider" thing can get interesting. In this case Apple just hands your DMCA notice to the alleged infringer.

Apple is hosting user created content though (in the form of apps). Don't they become liable for that hosting if they don't handle the DMCA request themselves?

Yes, in this case Apple would be the recipient. And even though they did not create the infringing app, the DMCA take-down forces them to block access to it.

In effect, you'd be triggering the same sequence that led to youtube-dl being blocked on GitHub for a while. Surely, GitHub = Microsoft didn't build it, but they still had to block access when the RIAA sent a DMCA.

In the youtube-dl case, the DMCA notice was later found to be fraudulent and thus the access was restored. But if the infringing app re-uses images verbatim, that won't happen so it stays offline.