Can Medicaid take what's left in my Mom's checking account after she passes?

No. Withdraw the money and close the account.

In our case, in New York state, any money in a joint account belonged half to the Medicaid recipient and half to the joint account holder. We used her half (and more) to pay for the funeral and Medicaid was satisfied with that.

This might be a question your Medicaid case worker can answer for you. The rules can vary by state.

Yes, unless you can prove what you contributed to the joint account.