Can I apply for an annual extension after receiving a 7-day denied visa?

It's not a visa, it's the denial of one rather. That matters a lot: you don't have the right to stay in Thailand, you must leave. And you get 7 days to do so (not actually 7 though; the day you get the stamp also counts, so even if you get it on the very last day, it extends your allowed stay by only 6).

Think about what a problem it would be if they'd still allow visa applications during those days. If that'd be allowed, some would still fail to fulfill the requirements. Those would be required to still leave. How many days would they then give them? 7 again? Then there's no end. So such a "get out" stamp can only be final.

Nope you cannot. That 7 day stamp is actually an "application for extension not approved" stamp <- means you get a week to leave the country.

You can't get that and then during that 7 days apply for the extension sorry 🙁 Once you get that stamp you're done doing anything visa/extension-wise inside thailand, you got until that stamp runs out to leave thailand.

What normally happens is you just run on overstay until you can get the document you need for the extension and pay 500 baht a day. <- more than a few people end up doing that because they need some document or other..