The rules for Social Security Disability Insurance benefits and for Medi-Cal, which is California's Medicaid program, are completely different, so let's take them one at a time.
your brother's Social Security Disability benefits would not be affected at all by his inheritance from your mother's house, whether he keeps the money himself or gives it away to you. His Social Security Disability benefits are based on his ability to work. Social Security looks at how much money he earns from current work he might do, but it does not consider how much money he has in the bank or any money he receives from any source -- including an inheritance -- other than work.
Medi-Cal medical coverage is different from Social Security benefits. Medi-Cal is only for people who have very low income and also have few assets (usually only around $2,000, not including a home you live in). So, if your brother actually receives his $10,000 inheritance from the sale of your mother's house, it could disqualify him from Medi-Cal medical coverage until he spends his assets down to the eligibility limit of $2,000 total. However, if he gives that inheritance money to you and never actually collects the $10,000, Medi-Cal would not consider it as his asset and it would not disqualify him from coverage. You are right to suggest that the rules would be different if he were in a nursing home or would soon enter one and apply for Medi-Cal nursing home coverage (which is different from medical care coverage). But since that's not the case, he doesn't have to worry -- there are no Medi-Cal medical coverage rules regarding giving away assets.