I was a bicycle messenger there for years. As pointed out, most accidents are your own fault. I hit a bus, hit a truck, hit pedestrians, hit taxis, hit curbs and broke ribs! Traffic is major, but that means actual traffic speeds are slow. Manhattan IS very different, much more random and faster. Major city cycling is a very different kind of cycling.
Fold-up bikes are allowed on the subways, and on all 'overground' trains. I think on Sundays any bikes are allowed on certain subways too(check with a guard).
Don't take your eyes of your bike (even locked) Just watch for everything, go slow, don't get out of your depth, avoid pedestrians etc you'll have a huge amount of fun!
jon