To answer all of your questions:
As long as you stay in Manhattan south of 110th Street, there are no remotely dangerous areas. Everyplace is safe, and has been for twenty years, since Mayor Guiliani undertook a vast cleanup campagne of the city starting in 1993. So you don't have to worry about being in danger. There are some unsafe areas on the outskirts of Brooklyn, Queens, and the Bronx, but there aren't really any touristy things to do there, and you won't accidentally go to them, because Manhattan's an island, so you'll know if you're leaving it.
Tips are absolutely essential in the U.S. for restaurants and bars, along with luggage porters and taxi drivers. Be aware that waiters/waitresses and bartenders are usually paid extremely poorly -- by law, they're exempt from minimum wage requirements, and so almost all of them get paid about $5.50 (roughly three Pounds) per hour without tips, and so almost all of their income comes from tips. It's customary to tip between 15% and 20% of your bill -- if you're ordering drinks at a bar, it's acceptable to just tip a dollar a drink.
Corn Dogs and sloppy joes, I don't really know, but if you're staying at the Hotel Pennsylvania (which is a very popular tourist hotel across from Macy's, so I assume that's where you are), there are plenty of touristy delis in the area which should have whatever kind of food you're looking for.
Most restaurants are open until 1:00 a.m. or so; there are plenty of places open all night -- near your hotel, there's a decent diner on the corner of 34th Street and 8th Avenue that's open 24 hours. Bars close at 4:00 a.m.
No particularly unusual local customs; we're not that different really from the UK. One thing that annoys New Yorkers, although we're used to it, is when tourists stop in the middle of the sidewalk to stare up at all the tall buildings -- it gets in the way of people trying to walk places. just to be courteous, it's nice to step to either edge of the sidewalk (near the buildings or the stretet) to let people pass in the middle.
Macy's is a department store that sells lots of different things, mostly clothes. Think of it like Harrod's.
McDonalds are everywhere. I can think of three within one block of your hotel.
As for fun parks and places to go, there are hundreds of them, too many to mention.
Have fun, and enjoy your stay!