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It does look like a palo verde, and these trees can show invasive behaviors and spread from the roots (probably part of a desert survival strategy). If you can somehow cover the area where they sprout with an opaque covering to shut out light, the shoots should at least stop growing, and hopefully the roots will eventually decide that area isn't a promising direction to grow in.

These articles are about other types of trees but may help:
http://diagnostics.montana.edu/plant/topics/treeshoots.html
https://www.gardeningknowhow.com/ornamental/trees/tgen/tree-sucker-removal-and-tree-sucker-control.htm

This can depend greatly on the type of tree that you are referencing.
Some trees can recover and do well after a rejuvenation prune.
The correct time of year to prune is very important.
Tree pruning for many varieties is done when the tree is dormant.

You can research your tree type on our website for more exact pruning information.

I do not know why the photos posted sideways-- they were right side up when I uploaded-- my apologies.

https://www.gardeningknowhow.com/ornamental/trees/tgen/trees-with-invasive-roots.htm

The tree might need to be removed. Cutting off the roots on one side of a tree can sometimes destabilize it and cause it to eventually fall. This could be dangerous, so it is best to ask a local arborist or other tree expert to come look and help you decide whether the tree needs to be removed.

In other cases, the tree can be saved by making some changes. See this article:

http://aces.nmsu.edu/ces/yard/2008/091108.html

The pruning spray is unlikely to kill the tree, but it is also unlikely to be an effective treatment for tree wounds. Generally, it is best to let tree wounds heal by themselves (with the exception of a few specific types of tree). The pruning spray porbably won't get deep enough inside the bark to inhibit nutrient transport.

One problem it could potentially cause is overheating if the tree is in bright sun. Perhaps next year the black spray will wear off, or you could whitewash over it.

If the deer removed bark in a ring all the way around the tree, known as girdling, the tree will die anyway. The only solution is to do a complex grafting procedure:
https://www.gardeningknowhow.com/ornamental/trees/tgen/how-to-fix-girdled-trees.htm
If not, the tree will probably recover.

Yes, it is fine to use lava rocks or river rocks instead of mulch over tree roots.