Can the executor of a trust move accounts out of the trust that are designated to other people?

The trustee in a trust must make an accounting to the beneficiaries when they request that she do so. You should put this request in writing. The monies should remain in the main trust, unless the trust gives the trustee (your mother) the right to move the money around. Caveat: If your mother is the actually beneficiary as well as trustee, and you don't get money until her death, she may be able to do what she wants with the money. It all depends upon what the trust says.

Obtain yourself a probate litigation attorney NOW to bring a petition for an accounting of the assets of the trust, and to whom paid, and when paid, and have removed as executor/trustee from the trust and to force her to produce the assets to have the assets placed back into trust. She may have committed embezzlement, which is re-portable to the District Attorney for prosecution. This is urgent.

Probably not but it depends upon the wording of the trust. You should consult an estate planning attorney to review the trust for you and advise you how to proceed.

Probably not. You are using terms loosely. There is no such thing as an executor of a trust. Estates have executors. Trusts have trustees. They are not the same. Take all the papers you have, and your questions, and go see a lawyer.