Can I continue using my iCloud email address without connecting to the Cloud?

Short version, no.

Slightly longer version, Apple's Mail services uses the IMAP and SMTP mail protocols. These require mail to be received, stored and sent from a server - in this case servers owned and hosted by Apple. You can certainly save copies of mail locally on your computer (and then delete them from the server) but you can't get around having to interact with the server, and at some point every email you send or receive goes to "the cloud".

This is no different than Microsoft Live, Gmail or a host of other mail services. There is always a server in the cloud (i.e., on the Internet) involved. The only way to avoid that would be to host your own email service, with your own domain and a DNS service record published so that other mail servers can find yours.