Yes, you can run a Freesat box from a Sky dish. The two services use the same satellites and mostly the same broadcasts from those satellites. I have run a Freesat box off my own Sky dish without any issues.
The connection between the dish and the LNB ... yes, with an adaptor clip. Original Sky minidishes apparently have a non-standard mount for the LNB. However Maplin sells the adaptor, and presumably other retailers do too.
http://www.maplin.co.uk/module.aspx?moduleno=221062
