One big thick limb of a choke cherry tree has decayed. It and all attached branches have fallen to the ground. A black, red and white Woodpecker was pecking away in the decaying wood. Should I remove the affected broken area. Or, shall I have the entire tree cut down?
It is only necessary to remove the branch that is dead, unless you do not want the shrub, anymore.
I assume, since you mention a thick limb, you mean Prunus viginiana and not Arronia. I will include an article for this:
https://www.gardeningknowhow.com/ornamental/trees/chokecherry/chokecherries-in-the-landscape.htm
My choke cherry tree had a large mushroom at the base of the tree last summer. The trunk also has cracks and sap. However, it did produce fruit and very large flowers. Is the tree dying? We are new residents and unfamiliar with the tree. Thank you.
This will depend, heavily, on the mushroom in question, and whether the tree has a disease unrelated to the mushroom, or if the mushroom caused it. Unfortunately, your photos did not come through, so I am unable to give you a detailed diagnoses. Something like a Reishi will cause the death of a tree, where as most Oysters or Omphalotus, Chestnut mushrooms, etc, will grow on trees, but not damage them. They are only taking advantage of damaged wood.
This article will go over some of the most common mushrooms that will destroy trees:
https://www.gardeningknowhow.com/ornamental/fungus-lichen/types-of-tree-fungus.htm
This one might be useful:
https://www.gardeningknowhow.com/edible/fruits/cherry/treating-armillaria-rot-of-cherries.htm