We did Lombardi’s twice, and I would definitely recommend it, some say it was the first pizzeria in the US.
Sorry, but for #3, while NYC is know for its great pizza, there are many mediocre and just plain bad pizza places (mostly slice bars). I would not recommend just trying any place off the street to grab slice. Like TJ Jackson said, go to sliceny.com and look around. If it’s your first time in Manhattan, if you want to go for an "old school" New York pizza parlor, check out Lombardi’s, Patsy’s, or John’s. I can’t vouch for Patsy’s, but the pizza at Lombardi’s and John’s are both good, though better may be found elsewhere in Manhattan, many lack the history of either.
As for another New York classic, pastrami, you can’t go wrong at either Carnegie Deli or Katz’s. Both serve excellent pastrami, with the rye bread being better at Katz’s. Carnegie gives you more bang for your buck as you get more pastrami for slightly less money and in my opinion serves a better meat than does Katz’s. Both have great atmospheres. I’m partial to Carnegie, but if you have time I recommend sampling both (if you do Katz’s also makes a good hot dog – ask for it well done).