There is no law prohibiting it. However, some VFW's are anti-gun and will eject you if they discover you are carrying. There are a couple of threads on the subject floating around here somewhere and I'll post them if I can locate them.
Yes, depending on the rules of the individual AL post. Generally this isn't posted online, but a telephone call or a stop by the post will answer your question.
That is per AL HQ in Indy when I called earlier this fall.