You've got a whole lot of questions here, but I'll try to break it down as best I can:
1) There are some hostel options in the city, but be aware that 50/night is still a pretty tough target to hit. NYC is a very expensive city, and nothing is more overpriced (relative to the rest of the country) than real estate - which in turn, makes turning a building into a hotel or hostel much more expensive than anywhere else, which in turn, keeps rates high.
2) First off, Jamaica isn't the suburbs. It's smack dab in the center of Queens, which is one of the five boros of NYC, and very much a part of the city. Secondly, unless you have family or friends out in Queens to stay with or something, I'd advise against doing this. It's not a neighborhood that caters to tourists, and you'll end up spending a lot of time riding the train for a pretty minimal savings (if any at all). Really, the only good reasons to do this are if your primary destinations are in Queens (say, if you were coming to town for the US Open), if you have a car and the savings in parking dramatically swing the overall cost, or if, as I mentioned, you have someone to stay with that reduces your cost of accommodation is free.
3) A 7 day unlimited Metro Card is 30 bucks. Adjusted for the 5% bonus, this is the cost of 12.6 rides. If, in the course of your time in NY, you'll be getting on mass transit more than 12 times, it's a good deal. If you think you'll be using the system, say, less than 8 times, it's an objectively bad deal. If you think you'll use the system between 8 and 12 times, the added flexibility of not having to worry about it might be a good idea.
4) Express Busses are primarily commuter routes that connect the furthest reaches of the city with downtown destinations while skipping a large number of stops. They cost more. An Express Pass allows you access to these busses. You do not need one as a tourist.
5) Go to a PATH stain and get on a PATH train. You'll be able to use dollars stored on your metro card (but not your unlimited pass) to pay the fare, which is 2.50 each way.
6) AMTRAK is probably your best option here, though if you're really trying to save money, there are dozens of low cost bus companies that run along the I-95 corridor between DC, Baltimore, Philly, NYC, and Boston. Use Google.