Is the Zakat sufficient if the mother pays its value from her own money after receiving the amount sent in dollars due to her need for it, and must the Zakat payer pay the Zakat again in this case?

If your mother paid your Zakat (definition) to those who are deserving of it, with the intention of Zakat, then your Zakat is valid and you are not required to repeat it. It is not necessary to transfer the exact same dollars; rather, as long as she disbursed their value in your country's currency, there is no harm in that. What you paid to your brother with the intention of Zakat is sufficient if he was poor, because charity given to a relative is both charity and strengthening of kinship ties. The agent should avoid purchasing for himself in the future, due to the jurisprudential disagreement concerning this matter.