A friend ( woody plant expert) believes these are Japanese maples. I think you should be able to locate some leaves beneath them.

"When I take it out of the container, I noticed that its root actually is wrapped by burlap." Containerizing B&B stock isn't unusual. It's just a less-than-optimal way of holding nursery stock over. "The burlap already rot(ted away)." That may be true, but I would double check to make sure that is the case before planting. As Ken said, I would also remove the burlap showing around the base of the trunk. Typically, the burlap, in this type of scenario, will be embedded below the outer layer of soil. When the B&B'ed rootball is placed into the pot, soil is filled in around that, and then roots grow through the burlap into the surrounding soil. You may find that the burlap is still very much intact and just below the added soil's surface. Also, check the level at which the tree was planted. It looks like to me that they've added lots of soil on top of the existing rootball. This excess soil needs to be removed down to the top of the original rootball. Otherwise, the tree may end up being planted too deeply. "from the pics, do you think there is rotten root problem?" No, that part looks fine to me. I see no evidence or rot. "should I cut those roots out of the burlap?" Maybe, but this question is somewhat confusing to me since you said the burlap had decayed. Is it there or is it not? "should I treat the tree?" If by treat, you mean address the problems I mentioned above, then yes. Otherwise, I'd need to know what you meant by treat.

So. Does this look like an Amur Maple? ... finally has leaves out which were not damaged by the freak Artic Spring blast this year. The flower bud cluster has formed, but not open yet. Here is the multi-trunk, this would be very hard to remove without machinery I should think, but I'm am considering it... the surrounding soil is full of small roots also, very hard to even dig any hole more than a couple of inches

The reason the 1/2 gal of water came out the bottom is because the soil was very dry. It just ran through and not much of it probably stayed in the soil. It's amazing how little moisture really dry soil will absorb before it runs through the pot. Give the plant a shower. Just a short one. The plant will take in water that way. I've revived dry plants just doing that. Just keep adding water a little at a time. Give it a chance to rehydrate that soil. Stick your finger down into the soil. You'll be able to judge how much moisture your soil has absorbed. You'll only drown your plant if you let it sit in a pan of water a long time, like a day or so. Give it an hour or two at least.