I think that having a private exit node pool would defeat the purpose. To make this pool private, it would have to have some sort of security -- such as a password, or security by obscurity. Either way, the user would have to know something about the node(s) and would therefore reduce the amount of anonymity.
In reply to the other idea, I think that may be possible, but it'd probably have to be a https request on port 443, and because TOR is a tunnel, I'm not sure if it would get to the right port at the end. Each packet sent through TOR has a header with instructions to take it through the hops in the tunnel. Any single node can only read its own part of the header, so that particular node can't tell where the packet came from or where it is going. If you knew about the private exit node, then the hops in the middle would be pointless, because where the traffic was exiting would be known. At least, that is my theory.