Does Time Capsule require Internet to properly function as a backup server?
Time Capsule does not require an active internet connection to function correctly. I would, however, caution that these units are notorious for getting 'unhappy' when used in an area that does not have sufficient cooling. Putting one inside the drywall is a bad idea, IMO. Unless you're using a wired connection to connect to your TC, then you're either going to need to set the TC up to extend your existing wireless network or you'll need to have it on its own network. That latter makes no sense, as your Air will not simultaneously connect to multiple wireless networks. To make Time Machine work as intended (hourly, autonomous backups), you need your TC on the same network as you use for surfing and working.

A TC will quite happily operate without creating a wireless network (Wireless | Network Mode 'Off'). I'd suggest that your ideal configuration would be to have the TC connected with Ethernet to your existing wireless AP running in bridge mode. The TC would get an IP address from the DHCP server in your router. You can then configure the share in the TC and point Time Machine from your Mac to use the share.