Does workerd have full feature-parity with Workers, including stuff like Durable Objects?

It's the same code. The only thing it doesn't include is the global scheduling and orchestration, but you don't actually want that locally. (And honestly... that stuff is a bit of a beast.)

Durable Objects are fully supported by workerd (and Cloudflare OS uses them extensively).

There is, however, one catch currently: Durable Objects don't scale out well without the global scheduling. For running Cloudflare OS for one user, this is actually no big deal, but a company-wide instance might not work well.

But I'm actually fixing that:

https://github.com/cloudflare/workerd/pull/6780

Unfortunately didn't get enough time to land it before this release, but it'll come soon.

With that I would expect a self-hosted Cloudflare OS on your own cluster to work quite well.