Hi, Bonny!
As you no doubt know, my experience with mules is limited to one, the mule of my heart (and my avatar,) Betsy. My answers are based on what I have observed with her.
Yes, mules shed. When they do it for the first time, I haven't got a clue. I'd expect it at about the age a horse foal would do it, if the weather doesn't interfere. I think Molly ought to start, at least on her face, pretty much any time now. Betsy's winter fur is pretty long and soft, almost stuffed-animal cuddly. It's so nice to be able to hug on her, and not have her get all tense any more!
Betsy's summer coat is very slick, short and shiny. I'm told that horse hair coats tend to be "hard" enough to shed water, at least enough that the animal doesn't soak to the skin in a good rain. Donkeys, on the other hand, have a softer coat, and they will wet all the way down. I don't know if that's so, it's just what I've heard. I don't think Betsy is as hard-coated as Sydney (the mini) is, but she sheds rain pretty well. On the occasions that I have checked, she wasn't wet to the skin, but she does have enough sense to head for shelter in a really hard rain.
We don't have any "boys" around here, of either species, so I don't get to see full-fledged heat behavior regularly from any of my girls. I have seen Betsy flirting with geldings a few times, and she really liked a mini stallion that was stalled near her at the fair a couple of years ago. She lets Syd (who is a filly) mount her sometimes when they are playing, but I don't know if this says more about Betsy's hormones, or Sydney's!
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