If the house sold would my dad keep the profits?

You have to report the proceeds to Medicaid IF the house was in BOTH names. Talk it over with mom's caseworker IF she is on Medicaid.

Sounds like your Mom is on Medicaid and Medicaid is paying for the nursing home. If that correct? If yes, depending on your State, Medicaid could claim half the profits. There are exceptions to the rules, you would need to contact Missouri's State Medicaid office.