Macbook air A1370 use battery A1406 Macbook air A1465 use battery A1495 they are not cross compatible, if you use the wrong battery your computer will become extremely slow.
Reflowing the board is a waste of time. You can't fix a bad GPU by getting it reflow. you need to replace it. By the age of the computer i feel like is better for you invest in a new macbook. Because that computer had a known GPU issue but apple end the GPU recall on that model already. Good luck!
Hi, The bottom case of the macbook already have your serial number. If you buy a third party bottom case it will either come with a random serial number or with no serial number. Hope this help. Good Luck!