Hi Hermione!
Wow, you've been planning well in advance

Good for you!
It sounds like your 17th is going to be a blast! It's nice that you'll have a more intimate party for your 16th. It's going to be a memorable year!
Laser Tag and Rock Climbing both sound like a great idea for you and your friends. Other ideas could be paintball, kayaking, mini-golf or the driving range (may sound silly but can be really fun).
The Long Island Tourism website has a few venues you might like in the amusement parks section. There is a Boomer's by us and it is pretty popular for teens, there is one in Medford, LI.
As far as restaurants go, there's so much to choose from. Do you know what kind of atmosphere or cuisine you're interested in having?
Good luck and Happy Birthday!!!