Is the Eid prayer valid if the first rak'ah has three takbeers and the second has three takbeers before rukoo'? If it is not valid, must it be repeated, and when?

The Eid prayer you mentioned follows the Hanafi school of thought. In "Al-Muheet Al-Burhani" and "Bada'i' Al-Sana'i'", the Imam pronounces nine takbirat (glorifications of God) in the two Eid prayers: three original ones (the opening takbir and the two takbirat for bowing), and six additional ones (three in the first rak'ah and three in the second). The takbirat precede the recitation in the first rak'ah, and the recitation precedes the takbirat in the second rak'ah. Based on this, your prayer behind this Imam is valid.