If his nose is a variation of liver and not actually black (I thought it was black from the pics) then yeah he could be purebred.
After reading all the stories of his shenanigans swiping your stuff... it wiped any doubts of his beezerness from my mind.
I definitely don't think azawakh. Aside from being incredibly rare, their behavior is soooo different from other sight hounds. They're used as guard dogs in their homeland, most in the US are less than 6 generations out of working for a living. They're not just "aloof", they're downright distrustful and I'd expect an azawakh mix to be a lot more suspicious of strangers (they're much more like a cur in that way)
If he's mixed, I'd think it was podengo. Plenty of beezer breeders keep podengos on the premises so if there was going to be an oops it's more likely it would be something like that.