You are allowed to bring your bike on the subway at any time. If you want to be courteous, try not to bring your bike on the train during rush hour, and try to be at the front or the rear of the train, and towards the either end of the car so you aren't blocking people. But NYC is getting bike-friendly enough that you're probably better off riding wherever it is you need to go (it's also the best way to explore the city) than taking the subway.
As for a place to stay, I've had a good experience renting an apartment from craigslist for a few days. It was considerably cheaper than a hotel. It was also illegal. (
http://nymag.com/realestate/realesta...ntals-2011-12/)
Good luck, and enjoy our fair city!