Absolutely you, your family, and your pets can catch sarcoptic mange from this stray dog. In your shoes I would bath all the pets showing symptoms in Ted's Mange Remedy. It is critical that you decontaminate your house, so you need to clean and treat all the areas that your pets frequent inside; it may help to put down borax to decontaminate areas.
If you are now feeding this dog, and have some of his trust, can you possibly catch him? I believe he is curable, provided you can bathe him. If putting hands on him regularly is out of the question, I would consider Ivermectin/Ivomec - an oral miticide - in his food. You can purchase Ivomec online.