Hello Jack,

The die back of branches may be caused by girdling roots below the soil surface. A certified arborist could examine the tree and determine if they can be removed without endangering the stability of the tree.

https://yardandgarden.extension.iastate.edu/how-to/all-about-stem-girdling-roots

The twigs coming down are likely from squirrels(the broken end is often at a sharp 45 degree angle) or wind damage( broken ends are very ragged and random). This type loss isn’t harmful to the tree. Take a close up picture of a couple of the broken twigs and add them here so we can see them, if you would like.( see picture from Ohio State University Extension below, of squirrel damaged twigs)

https://apps.extension.umn.edu/garden/diagnose/plant/deciduous/elm/leavesground.html

The green growths aren’t coming through on the pictures but, they sound like lichens, which come in many colors and do no harm to the tree either. https://www.canr.msu.edu/news/what_are_lichens_and_what_are_they_doing_on_my_tree

What you can do to help the tree is to remove the sod around it( in a circle or an oval or a square shape- it doesn’t matter) and put 2-3 inches of mulch over the bare soil. This reduces competition for moisture and makes it easier to mow the lawn around the trunk without injuring the bark. Never pile mulch against the trunk as this encourages rot.

To find a certified arborist to check the overall health, check/correct girdling roots, search by zip code here- http://www.treesaregood.org