Ask Mr. Smarty Plants

The very first thing you need to consider is whether there was any damage to your trees during construction. Since this is out of our area of expertise, and we don't know what degree of leveling is going to be necessary, we suggest you get a landscape specialist in to do the work. Insist on first checking the property, especially around the trees, to see what damage has been done by trucks parking over roots or supplies stacked there. Compaction of soil over tree roots can be deadly; see this article from Colorado State University on Healthy Roots and Healthy Trees. 

Now that we've answered a question you didn't ask, we'll get back to the one you did ask, about  invasive vines and root suckers in your yard. There is good news and  bad news; first the good news-all those vines are native to this area. Now the bad news-that doesn't mean you have to like them. This is a good time to begin attacking the problem because they are deciduous and starting to drop their leaves, but since they are native perennials and can easily be replanted by birds dropping seeds they have picked up somewhere else, you will never finish the job. Besides getting the unwanted vines off your trees and the ground, you need to cut them at the root. Get some wide spectrum herbicide and small disposable sponge paintbrushes. As quickly as you cut a plant off at the root, paint the raw cut edge with undiluted herbicide. It has to be done right away to help make sure the herbicide gets to the root before the root starts healing over to protect itself. Be very careful to avoid spilling the herbicide on the soil or getting it on a plant that you want to keep. No spraying!

On to the root suckers from the trees. According to this article When tree roots sucker, it's usually in response to a wound, by Dennis Hill of Cox News Service,  suckering may be a characteristic of a particular species of tree, but usually it is in response to some sort of damage to the tree. If you cut a tree down, you will get suckers as the tree root struggles to survive. Without upper branches, it needs those suckers to quickly grow some leaves and start making some food before the root's own supply runs out. Your trees haven't been cut down, but they may very well be suffering from damage, and they, too, are trying to survive. Don't use the herbicide on those suckers, they are still connected to the tree root, and you are trying to save that tree, not kill it. 

Landscape fabric: Since we have no personal experience with landscape fabric, especially not to control root suckers, we found this article by Linda Chalker-Scott, PhD, Extension Horticulturist at Washngton State University The Myth of Landscape Fabric. If you keep nipping off the suckers, and work to continue improving the health of the trees overall, we think the problem will eventually grow smaller, if not go away altogether.