What are some solutions for remotely connecting to an Android phone?
Yes, rooting is usually (but not always) required. The following are some options:


Android Screencast: Probably one of the more easier and popular sol'n
Google Code Project "Android VNC". Have to search for it, I can only post 1 link at a time right now...
Droid Explorer: App from Marketplace
Android VNC Server: A guide that uses a different APK from either sol'n above


Try them out and see which one gives you the best options.