My Dad had a stroke and needs an adult living with him. I am buying a larger house using the equity from his paid-off condo to make it possible for me to live with him. My concern is that should something catastrophic occur that doesn't result in his death or just if his health declines enough that I can no longer care for him on my own, will I suddenly be forced to sell and be without a home even if I'm paying the balance on the mortgage?
If the home is titled in joint names, then as long as the value of his joint interest in the new home at least equals his equity in the house he sold, no gift is made. Then, if your care for your father (for at least the two-year period immediately preceding his entering the nursing home) enabled him to delay going to the home, he can legally transfer his interest in the home at that time, and there will be no transfer penalty. If you plan to go this route, you will need (i) a letter from a physician indicating that your father is in need of nursing-home-level care, and (ii) detailed records of what you do to care for him. Your state may have additional requirements you should find out about. The state will want this should your father later sign a deed of his interest over to you (after he enters the nursing home), to avoid characterizing the deed as a gift.
In my state, if a relative lives in the parent's house AND makes it possible for the parent to avoid Medicaid-funded housing for two years, the house is exempt from Medicaid reclamation processing.
If the house will be titled in your name, I have no sure idea how the condo equity plays in, but there are generally rules against transferring assets to a relative to avoid Medicaid reclamation. If you run afoul of those, the state will absolutely force you to sell.