I think you need to have a special permit to get lessons in you own aircraft.
I dont think so. For training the aircraft always use to have to be on a public cat but that may now have changed. For the IMCr there is no need to fit screens (and no one does) which is far more of an issue if you were doing an IR. The aircraft obviously needs to me on a European reg. (various issues arise if you want to use an N reg) and perhaps there are some issues if the aircraft is not on the G reg - that I dont know.