Yup, weeping mulberry. When I was a kid growing up in CT, these were fairly common in front yards. Sometime in winter they would be cut back hard, in summer new growth would cover the old limbs. In fact, the were pollarded, all though I'm sure the owners didn't know that was what the were doing. You can do the same to your tree, make it smaller, and keep it that size indefinitely. Google the term, look at the images and see if you like it. I personally don't, at least from what I remember as a kid. These trees are grafted, so what you are seeing are shoots from the understock, which would be a full sized non weeping tree.

I think it is a nice tree, definitely not too close to your house. Correctly identified.

Yeah I was reading about the grafting thing during a break today. Looks like weeping is usually grafted onto a white mulberry. So if i transplant one of the suckers I'll probably just get a white mulberry. Could transplant and attempt to graft off the weeping mulberry to get a new one I guess. But I'm no grafting expert though.

On the distance from the house thing. I thought mulberry was one of the do not plant near foundation trees, that has very aggressive roots. Does the stunted "weeping" top work to also stunt the roots. Shouldn't I be concerned that it's putting roots up into the crawlspace. To me it seems sort of alarming that the roots have already breach underneath the footer and come up on the other side.