You need to stop posting on this question and adding more wrinkles. I'm sure we've told you to seek professional advice. No one on this board will represent you at an audit, or will pay your Dad's medical bills if you guessed wrong on the MediCal lookback rules.
https://ttlc.intuit.com/community/home-loans/discussion/re-if-someone-refinances-their-primary-resid...
Generally speaking, the rental income belongs to the owner. Period. If you are the owner, you get the income.
Be aware that if your father gifts you the house, he also gifts you his cost basis. This is significantly worse for you than if you were to inherit it, especially with California taxes on capital gains. Also, when you place the property in service as a rental, the basis for rental depreciation is the current fair market value or your cost basis, whichever is lower. So the gifting strategy may seriously affect your rental income.
Then, you will be getting in bed with a renter. Are you prepared to deal with the laws as regards to tenants' rights, discrimination, and so on? What if the renter doesn't pay? What if they trash the place? Do you know your rights and are you prepared to add full time landlord and maintenance man to your other responsibilities?
You need to see an Elder law specialist for your estate and tax planning needs. Tell them what you want to do (keep the house, avoid the Medi-cal lookback, and generate a little income) and let them tell you how best to do that.
