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Do they look like the acorns on this page?
http://www.notcot.com/archives/2013/08/gall-wasp.php
You might have knopper galls, a symptom of a certain kind of wasp. There are actually wasp larvae growing inside the acorns. It may look really weird, but the presence of the galls won't harm the tree.

This is my go-to field guide for oaks. I think you'll find it helpful.
https://www.fs.fed.us/foresthealth/technology/pdfs/fieldguide.pdf

The bark looks to have lichens growing which often aren't a problem. However, given the lack of health of your tree, it may indicate a weakened, sick tree. Red and live oaks are susceptible to oak wilt disease. That your tree didn't leaf out means it is a dying, if not already dead tree. Going into winter with no chance to renew its energy supply will kill it off. If you aren't convinced and since it needs some pruning, you can double check yourself. Cut off the lowest branch on the left (it is competing with the central leader) and look for signs of life just inside the bark. If the entire inside is dull tan, it is dead. Internal rot may keep branches from being brittle.

Apply a fungicide to the top and sides of the tree stump to prevent future mushroom infestation. Look for fungicides that contain the active ingredient triadimefon, borate salt or chlorothalonil for best results. Before applying the fungicide, read the package instructions carefully.

I'm sorry that your plant suffered damage.

Your plant can survive this accident.

If the trunk at the break point is less then 6inches in diameter it will branch at that point into several segments and keep growing.
If your plant is more mature, it will also split and continue growing. Ponytail Palm are very slow growing so recovery will be slow.