The short answer is yes as long it is a single family home and the executor is the trustee. Since it is an executor it would probably have to be refinanced as an investment home with the higher rate, the executor would need to take on the responsibility of the note, and it would be a bit more document intensive than a regular refinance.

The living trust has a trustee, not an executor. The will is a separate process and you would be the executor.

The executor must file an account that lists all the assets of the decedent both real and personal property. That means all bank and investment accounts, vehicles, furnishings, art, jewelry, etc. That inventory is public and you can request the probate file and examine it. If you think the executor has not done a thorough listing all the assets then complain to the court. The executor will later be required to file an accounting to the court detailing how all those assets were distributed.

What is the difference between an independent co-executor and a co-executor

Yes, the executor can be a beneficiary. The court may remove an executor at the request of the beneficiaries.