Yes, it has a local webui by default. You don't have to connect it to their cloud, though I do because it's free and lets me get notifications if something goes offline.

Correct, the discovery process where you add them doesn't actually involve the addon and works fine without it. The addon is just a wrapper for the standalone ESPhome web UI. It's also not the only way to author and flash a config to a device, you can do it from any computer and on the command line.

High availability storage is what the Ceph integration is for.

Edit: though it's kind of a pain to set up and probably way overkill. A separate TrueNAS or similar appliance with a 10 gig uplink will be easier and probably just as reliable for your use.