Depends on the terms of the will, and how the account was opened. You should consult with an attorney right away.

How can I tell if the executrix hired a particular lawyer for herself or for the estate?

generally the lawyer for the estate represents the executrix - it's synonymous.

How do I make sure I get a detailed accounting of my fathers estate from executor before it is final?. What should that include?

demand an accounting - if necessary hire a lawyer to bring a proceeding. and don't sign a release and refunding bond until you're satisfied.

How can I send an official request to the executor in a way that could prove the contents of my letter? Would a telegram do it?

Certified mail or other tracking would prove you sent something, it doesn't prove the content.

Can an executrix be legally removed from the duties, and another assigned?

Yes, but the correct term is administratrix. You should consult with an attorney right away.

What rights do I have to potentially pursue a case?

I agree with Mr. Popovich. Under the laws of intestacy you should be entitled to a portion of the case - and of the prior estate. you should...

easy - make sure the account requires two signatures to transact!

What does it mean?

i'm not sure what you are asking - what is an estate, or decedent?

Do children have a claim to real estate if there is a surviving spouse? (no will)

You should start with looking at the deeds, then read the intestecy statute. Or consult with a lawyer.

Is it a breach of fiduciary duty to neglect to pay taxes on a trust for ten consecutive years?

sounds like it to me. i'd hold them liable for interest and penalties.