ANew York City: Most convenient buildings to rent apartment in Brooklyn under 2750$/month?
Have you searched HomeAway.com? They offer monthly short-term rentals that might fit your price range. http://www.homeaway.com/search/new-york/brooklyn/region:5166/maxPrice/675/currency/USD
APortland: Cheap accommodation for one person
Airbnb shared rooms or CouchSurfing would probably work for you. Taking a Bolt Bus between cities is probably the cheapest option for transportation. It only cost me $17 to go from Portland, OR to Seattle, WA vs $50 via train. Although...