Is my prayer valid after I caught the prostration in the second rak'ah of the Eid prayer, and I prayed one rak'ah alone without the extra takbeers and without repeating the tashahhud? And do I have to attend the Friday prayer if I missed the congregational Eid prayer on the same day?

Whoever catches the last tashahhud in the Eid prayer with the Imam, but misses the first rak'ah, must perform the two rak'ahs of Eid prayer after the Imam's taslim, including the extra takbirat, otherwise the prayer is missed. Scholars differ concerning making up the Eid prayer if it is missed, and the correct opinion from the Shafi'i madhhab is that it is always recommended to make it up. As for someone who did not pray the Eid prayer with the Imam, the Friday prayer is not waived for them.