So much of this is a matter of opinion. Museums, for example. It really depends on your taste. I can tell you that the Museum of Modern Art is the truly hot museum right now, because it recently reopened after a renovation. It looks great.
I’m not much of a deli guy (blasphemy, I know, especially for a Jew such as me), but while people I trust are still very keen on the Carnegie, Katz is the one they feel is superior. I’d personally go to Carnegie specifically because of the greater, non-deli type options, but I’m a goofball. I was just at the Carnegie last week, actually.
I do love Italian food, though, and don’t think so much of Little Italy beyond the sheer interest of seeing it. There are a lot of great little Italian places around the city that I think are great. One I haven’t tried but hear very good things is on the Upper West Side and called Nonni. I heard about it most recently from a former Roadfood poster. I want to preserve his anonymity, so I’ll refer to him (or her) as I-96.
As far as other items on South Hillbilly’s list: I LOVE Balthazaar, and in fact think everything else you mentioned is great (I’ll take your word on the tea at Payard). If you’re up for another "nice" meal along the lines of Balthazaar, one of my absolute favorites is a restaurant called Artisanal, on Park Avenue South and 32nd Street. The specialties are cheese and wine: literally hundreds of each. I’ve made entire meals out of cheese and wine, with fondue for dessert. If you like cheese, it’s amazing. I suggest reservations, though.