Flushing is like Little Asia. I flew in the Far East for 7 years and if you dropped me in Flushing and said I was in HKG, I'd believe you. Don't' know about rental prices though. That said, checkout Astoria for close proximity to Manhattan w/o Manhattan prices. Its next to LGA and has a young, cool vibe with lots of diversity and restaurants, etc. Bayside is a little more "neighborhood-like", but tons of rentals.

All are convenient to JFK/LGA.

If you use a broker to find a place, they may try to charge you a broker fee of up to 15% of the whole years rent! Tell them you'll only pay one month at the most and if they tell you no, find another broker. There are thousands. I know b/c I was one. Fee's for brokers are negotiable 10000% despite what they tell you.