Proper american english would be "May I open the...." Is this not the same in Italian or do they say "can...."
It depends. Maybe they're not asking if it's okay to open the window, but if the window is able to be opened? :)
it sounds like "posTo aprire la finestra?". It makes no sense of course, the slow version is more understandable.
che tagliare il formaggio?
What about 'I can open the window?' ?