I suspect that if you're getting different tracking results with the cover up or down that the problem is due to static electricity.
If the deck becomes charged, which is always a problem with Rega derived decks as they've no seperate earth, then with the lid down the arm is operating in an electric field. My P25 was a b****r on a hot day if I was wearing long sleeved shirts; a quick swipe of a cuff over the lid and 'boing' up would come the arm...
As a result I always use my turntable with the covers removed completely.
The siting of the deck on a piano is not likely to be a huge problem in itself. Though turntables do like to be as well isolated as possible, have a search through this forum and you'll find plenty of ideas! Keeping it level (as you note) is very important too.
I'm not convinced that this is an alignment problem; though that's not a problem I've experienced either. I've heard that cartridge in both Rega P2 and Goldring GR1 decks and although it seems very bright and shiny I don't think it would mistrack like that if it was set up properly.
It's worth checking the arm for free movement at the end of its travel too. I'd run the cartridge over Seb's 'Rega-Baerwald Protractor' (from the library section) and see how well you're doing for alignment. And I agree with Jas about running it towards the top of it's tracking weight range.
Hope this helps