Another possibility might be Rat Terrier. They do come in brindle, although that colour is out of favour. There is one in my neighbourhood that has a bit of a Basenji look to him, and his tail is very short.......not sure if it's natural or docked.
Looks like he is doing great at the dog park and with the other dogs - nice pictures, all of them! Gotta love the lhasa in the bright red coat - aren't we all thrilled we don't have to keep up with hair like that?!?