Yes, at 4 years old, he is eligible to enter Pre-K. Application requirements and registration processes for Pre-Ks (public and otherwise) are different for different districts and states. Make sure you check with the particular Pre-K that you're interested in to find out what their requirements are.
Absolutely yes. I don't think a child's preschool admission has any thing to do with his /her the grandparent's medical situation.
If you are thinking of what portion of the child care you may tax deduct or this particular school has any subsidy program that affects such decisions then it is different. I am not aware of those conditions. But in general your grand child is very ready to go to a preschool program and they should admit him provided there is an opening.