1) Maybe. Hopefully, but no guarantees. You are correct that the faster security line access doesn't get you through immigration & customs any faster.
2) Your choice. Especially with a short connection I'd take them with me to make sure the operating airline of the next flight has them.
3) Yes, there is a train which goes in that direction. Get on either the Howard Beach or Jamaica train; both will get you from T4 to T7 in 2 stops.