Could you publish the private key (after transferring BTC out of that address)?

OP_RETURN could be used for the domain name, and create a DNS on the blockchain.

Can that be done for the sending address as well?

Yes, but this would be on BTC's blockchain.

Also... this is more than http.  It's data.  You could register a database.

Put the name on the blockchain to locate the originating IP address.  The server at that IP would be queried for supporting protocols (http, etc.)

Perhaps the blockchain data could be formatted:
DNS://yourdomainname