Can a legal representative sue for something when they were not the representative?

I'm not sure what you mean by "legal representative." If a decedent dies with a will, the probate court appoints an executor. If the person dies intestate (without a will), the court appoints an administrator. Executors and administrators are called "personal representatives." If the parents had a living trust, their daughter would be called the trustee. If I understand correctly, you are asking whether the personal representative can sue for a breach of contract that occurred before she became the representative (that is before her parents died). The answer is yes. But the suit has to be filed within the statutory limitations period. The statute of limitations for breach of written contract is four years. So the personal representative can sue only for missed payment within four years immediately preceding the date the lawsuit is filed.

Daughter is probably collecting on the debt. There may be some state law protections, but you need to talk to an attorney in that state.

Yes, the legal representative can do what the deceased people could have done.

Yes they can.