Is what my mother does correct, that in the first Taslim of the prayer I say: "Peace be upon you and the mercy of Allah, O Allah, I ask You for Paradise," and in the second Taslim: "Peace be upon you and the mercy of Allah, O Allah, I seek refuge in You from the Fire"?

The Sunnah upon concluding the prayer is only to say the Tasleem, as the Prophet, peace and blessings be upon him, said: "Its consecration is the Takbeer and its deconsecration is the Tasleem." As for supplicating, asking for Paradise at the first Tasleem and seeking refuge from the Fire at the second Tasleem, this is not prescribed because there is no evidence for it. Supplication is prescribed after the Tashahhud and before the Tasleem.