Is my prayer valid if I perform Fajr and its Sunnah a few minutes before sunrise, and finish the Sunnah one minute after sunrise, and do I have to make up the Sunnah?

There is no need to repeat the Fajr sunnah (supererogatory prayer) after sunrise, as it was performed validly. Even if it were performed at the time of sunrise, it would be permissible to do so during a prohibited time (waqt al-nahy) because it is one of the prayers "with a cause." Ahmad chose to perform them (qada') during Duha (mid-morning), and he said that if one prays them after Fajr, it is sufficient. Ata, Ibn Jurayj, and Al-Shafi'i said that one should perform them as qada' afterwards, based on what was reported from Qays ibn Fahd that the Prophet, peace and blessings be upon him, approved his praying them after Fajr. This is also supported by the Prophet's silence, by analogy to performing the Dhuhr sunnah after Asr, and because it is a prayer with a cause, like the two rak'ahs of circumambulation (Tawaf). However, the majority of scholars hold that the Fajr sunnah should be performed first, then the obligatory prayer.