Will stomach acidity that comes up during sleep hurt my lung health as well?

Yes. There is a causal link between acid reflux and asthma. However, the reflux has to come up high enough to hit your vocal cords and go down your lungs. Some patients will sleep with the head of their beds elevating using cinder blocks to avoid this problem.
Possibly. If the stomach acid is aspirated (inhaled) into the lungs, then your lungs can be damaged by the acid. ..