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The best you can do without manual removal is to make sure it rots faster. Drill some holes in it and fill them with a high nitrogen fertilizer, then water the stump. Repeat this every few months and the tree stump should rot quickly.

These are called suckers. Most of the time they grow more shrub like than tree like. Beyond this, what you describe is something that can happen to grafted trees. The grated scion (the part above the ground) dies in severe cold. The more hardy rootstock (what the scion was grafted to) survives and since the scion is gone, sends up suckers to try to survive. What I mean to say is that it is unlikely the suckers at the base are the same variety of tree that you originally planted there.

Not too sure about the effects of sap on cars; however, the following article should be of some help: https://www.gardeningknowhow.com/ornamental/trees/tgen/how-to-remove-tree-sap.htm

Drill holes in the trunk stump and then pour high nitrogen fertilizer in the hole. The holes. The fertilizer will give a jump start to teh bacteria that help to break down wood.

They would not happen to be tent worms, would they?:
https://www.gardeningknowhow.com/problems/pests/tent-worms-tent-caterpillar-home-remedy.htm

The mold is likely lichen. It is not killing the tree, but rather a symptom of it dying. Lichen are especially attracted to trees that are sick and dying already.
What kind of tree is it and what sort of symptoms does it have? We may be able to diagnose it for you.

There are a few things it might be. The first is that it is dying of old age, especially if it's an ornamental. Ornamental trees have a rather short lifespan, 10-15 years normally. Another may be a fungus disease (it would not be visible like the lichen is). You can treat the tree with a fungicide, like neem oil, to help combat that. There may also be something attacking the roots, but on a large tree, it can be difficult to check the roots. Check around the trunk for any signs that there is damage there to indicate if there is a pest in the roots.