Looking for a good assisted living facility for someone who has very limited finances (SS and pension only -- less than $15,000 a year).

Is the person that is in need a veteran? You can check out Veterans homes and usually they are very reasonable. My in-law stayed in one and his family only paid under $2000 a month.

If they own a house, can the house be rented out  to produce income?

Have you applied for Medicaid? I would look into it. Here in my area, AL runs $3,800 to $10,000 a month, that is the base. My mother pays $4,800, my step father & his wife pay $6,200.