first of all, welcome to the site and congrats on your first snake! the runaway mouse should be fine to feed him. I doubt he picked up any nasty parasites or anything during the short time he was unaccounted for. as far as getting the stuck shed off his head, a good trick for the headshy ones is to soak him like you've been doing, but wrap him up in a washcloth before putting him in the water. that way his head is rubbing on it the whole time he's in there. this should do the trick. and since you've since figured out how to keep up the humidity now, you should be fine for the next shed as long as you keep it up.

i would guess that the mouse is a little larger than your friends snake is used to. i'd try again in a week with a smaller one. I've never heard of a blockage or anything that would stop the mouse from going down if it's the right size. your friend may have thought it was about the same size as before but just a little difference can make a huge difference! if that makes sense...