Using the Web IDE from Raspberry Pi

Do I have to still re-pair every time I take the battery out of the Puck?

I don't think so. You'll have to test though :)

Web Bluetooth when accessing IDE over http

Shouldn't be a problem - you don't need Web Bluetooth. That's the whole point - it all happens via Websockets.

has something changed in the online IDE very recently? I'm only getting Mac addresses now, not names.

On what platform? do you have a screenshot?

If you're using Web Bluetooth on the Pi I have no control over what's displayed - that connection windows is 100% in Chrome.