Mike, you don't have to have a registered public dns for seting up domain-name in asa, you can leave it bank and should not affect in any way asa primary role of permiting or blocking traffic. The settings can also be any name reflecting your internal private dns naming if you choose to do so.

domain-name for example needs to be configured for configuring ssh access to PIX when generating rsa keys as requires domain-name settings to be any name, at least this is the case for PIX versions 6.x and prior versions, not the case in versions 7.x domain-name is not required for generating rsa keys.

There may be other reasons where domain-name is required for something to function properly but I cannot think of any right now...again generally domain-name is not required.

Regards