No.I want to know if I can operate a 12v 2A solenoid from 4 penlight AA batteries
That depends on the coil resistance. Without knowing that this part of your question can not be answered. You state it's 12V @ 2A. That predicts (with poor accuracy) the coil resistance is 6Ω. Four fresh AA's with 1.56V each is 6.24VDC. 6.24 ÷ 6Ω = 1.04A. IF your batteries could maintain that power level for 1 hour then in 1 hour your battery would be dead. And no, it won't last for an hour. In short order it will drop out.what the current draw be will it have a negative impact on the life of the solenoid.
No.will it have a negative impact on the life of the solenoid.