So I saw the mod in action. Is there a way to edit the claims, so that everybody can open chests, use furnaces, etc. in the claimed area?

Yes, you can edit the default player config, which affects all claims for options that are not configurable by every player. You can choose which options are configurable by players. Check the main server config file.

Yes, but it looks like you need to be an admin (operator) and edit the "Default Player Config". This will effect all players though.

Or you can create a special "Server Claim" and modify it's permissions - but you need to be an admin again.