I have enterprise edition of Broadcast by Phone autodialer. I cannot schedule calls to run from my remote desktop, it states it cannot connect to the Gateway, however when I click start now they run.

This is because the call list is on the remote computer, and the gateway computer normally has no access to the files on the client/remote computer. To schedule broadcast, simply copy the call list file to the gateway computer, and schedule the call on the server machine.

If I run a Campaign from the remote desktop, I cannot shut down the remote computer or it will give shut down the outbound call campaign. Is this normal?

That is normal. The campaign/calls are sent from the remote computer whenever there are free lines available on the server. The server computer simply makes the call whenever a new call request is there.