No you do not. The next person in the list on your Grandmother's trust document steps in as the successor trustee. Powers of attorney are invalid once the principal has died. Your mother's Power of Attorney (which has expired) has nothing to do with your Grandmother's trust.
power of attorney, executor, or executrix
No, they are two different things. A trustee administers a trust and has a fudiciary duty to maintain it as specified by the trust. A power of attorney is someone that is representing another person.
A person can be appointed Power of Attorney by the individual or a court. Once they die, the PoA is no longer valid. Anyone can be named as an inheritor, even if they were the PoA.
Most states that I've looked up allow the person named as the attorney-in-fact in a power of attorney to register a car for the principal.
No, unless you have also been appointed guardian or trustee. Power of attorney just means you can execute certain acts for them (such as cashing checks, etc.) Even a legal guardian or trustee of an estate is only responsible for debts incurred by the named individual to the extent of the assets of said individual. They are not personally accountable to repay any debt where they are not a named joint debtor.
Yes. A person can name the same individual as their attorney in fact under a Power of Attorney and as the named Executor in their Last Will and Testament. Generally, a Power of Attorney is extinguished by law at the time of death of the principal. At that time the person named as Executor needs to petition the Probate Court for appointment as Executor.
Not legally. A power of attorney represents a living person. After their death, the will and its appointed executor are responsible.
Power of attorney has no effect on lifetime rights, except that now the person named in the power can exercise the rights in the name of the person who gave the power.
A patient has a Durable Power of Attorney only if they have executed such a legal document in the past. The person named in the Durable Power of Attorney is the attorney-in-fact. If an incapacitated patient did not execute such a document when they were legally capable then someone must be appointed as their guardian by a court.
While a living will typically does not carry with it a power of attorney, the general rule is that a person can name as many people to have a power of attorney as he/she wishes, but it would complicate matters to name more than one.