You will absolutely have an APO address at Al Dhafra. Shipping is no problem for most items, although a few retailers don't offer shipping to APO addresses. Surevapes does not, they are the only one I tried to order from that doesn't. The only thing you CAN'T ship APO is raw batteries, so if you have an APV stock up before you go. Also know that when you fly out, whether commercial or rotator, your batteries need to be either in a sealed device or taped over the ends.
Technically there is a restriction on shipping liquids via APO, but I have only heard of it being enforced for alcohol. I have had eliquid shipped multiple times since I've been in Afghanistan with no issues, and many people here get liquid body wash and other items all the time. Just make sure you are ordering from a retailer who packs it well so it doesn't leak.
APO mail does not go through any foreign customs. It goes straight from the US postal service, to the armed forces postal service, to you. Be aware that shipping times can be long and vary greatly. I am not sure about getting mail in the UAE, should be a bit faster for you since you are on an Air Base, but I have had packages take over 3 weeks to arrive. (I am on a small outstation in Afghanistan). All AE addresses ship from New York City so try to order from retailers near NYC for the fastest ship times.