Thanks for your question.
Yes, that's all perfectly legal and very common in fact. Solicitors are often witnesses to wills as it tends to help to ensure the Will is enforceable. They are also very commonly named as executors, sometimes but not always with another private person, so that they can ensure the proper and prompt administration of the estate.
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Solicitors who are not actually named as the executors can still be instructed by the executors to help in the administration of the estate -again this is very common.
If the children are named as beneficiaries then they have a right to see the estate accounts to ensure that the estate is administered correctly.
If they have doubts that the executor will administer the estate correctly, or might defraud the estate then they should write to the executor setting out their concerns (you can also ask about whether they consider there is a conflict of interest). If this is not dealt with satisfactorily then they can instruct a different solicitor to act on their behalf (with the other solicitor remaining as acting for the executor - so there are two firms involved in the probate) - this is what is called "contentious" probate. It can get quite expensive but it is sometimes necessary when parties have considerable doubts or concerns.
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