Doesn't sound like demodex
Demodectic mange starts with patches of hair loss - not a rash. The only time there is anything approaching what could be thought of as rash-like is if the dog damages his skin scratching, and even that is usually due to secondary issues. The hair loss typically starts on the face too, and may spread to the torso (usually upper) - not near the armpits. So no, while I can't say it's impossible, it
sounds nothing like demodex at all.
Demodex is not contagious either. ALL dogs have demodectic mites. So do you actually. D-mange is a symptom of an immune system problem, not the mere presence of mites.
More likely than that, if it's mange at all, is sarcoptic mange. That *is* contagious, including to humans. It is canine scabies (aka Nits). However, if the dog has had this problem for some time (has he?) and neither your other dog nor you have been infected - most likely the problem is something else. Sarcoptic mange does usually start out in the armpits though, and along with hair loss usually causes severe itching, pimple-like pustules, followed by skin traumas (actually usually from scratching). If you suspect it could be the case, it is best to see a vet for a skin scraping to find out.