| Crate or no crate, she is going to have to learn to stay alone. Leaving her in the mud room with the crate open didn't help. This sounds like a severe case of a puppy's normal aversion to being left alone. It is an unnatural situation dogs instinctively fear. If she came from a pet store, she may have been left alone long periods in a small cage making things worse. If you are using a wire crate, put it in a corner, under a table, cover it, etc. to make it more of a den rather than an exposed cage. If you bought a large one, perhaps pick up a small plastic one that fits her now. You could try a cardboard box inside a large wire crate to give her a comforting den.
Try putting her in the crate, closing the door, and lying down in front of it like you were going to sleep. Talk to her, perhaps singing to her. It has worked very well for us for many puppies the dreaded first nights in their new home.
I am slow to suggest medication, but have enough experience to know this could be a case for it. Talk to your vet. It is unnecessary in most cases. we have never had to use it. The techniques in Fred's link work with most dogs. Chances are, they were in a leaflet that came with the crate.
One more thing is a rack. I always use them with a new puppy until it isn't having any accidents in the crate. They are available with the crates, but a piece of closely spaced wire closet shelving from a home supply place is cheaper. The rack will allow urine and vomit to fall through and to a lessor extent stools. This makes it much easier to cope with the problem until the puppy adjusts.
Your ability to stop in and have the neighbor check should make the adjustment much easier. | 
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