While he will get BAH, sometimes it can take a while for pay to get set up correctly. It sounds like you don't have much time, but beef up the savings as much as you can. He will get back pay eventually, but that can take time.
And yes, they will pay to move the family possessions from his home of record (where he lived when he joined, basically) to his duty station. Military moves can be... challenging. I highly recommend joining this facebook group and reading all you can about the process. (Please don't be one of the may jackholes that says, "Hi, we are moving. What do I need to do that?" The info is there; search and read. Then, if you have specific questions, ask them. To get you started, there are 2 types of military moves--military-procured or PPM (personally procured move, which you will also see called by it's old name, "DITY"). In the former, a crew selected by the miltary shows up, packs, loads, drives, and unloads you. In the latter, you are given an incentive payment based on the weight you move (you provide weight tickets) and a few other factors. That's a set amount. It is NOT a reimbursement. If your incentive is $10,000 and you spend $3000, you get $10,000. If your incentive is $10,000 and you spend $12,000, you get $10,000.
For someone new, and especially because you spouse may not be there are may only be home on leave for a short time, I'd recommend a military-procured move as it's less paperwork you have to figure out. But you have to stay on top of them, watch them carefully, read everything carefully before signing, speak up if you feel they aren't packing something properly, and generally stand up for yourself. So movers are amazing. Others are... not. The more you know, the better you can advocate for youself.