Is a prayer valid if I catch the Imam in the last rak'ah, then pray the entire 'Isha prayer without counting the rak'ah I caught with the Imam?

If you joined the Imam in the last rak'ah of the 'Isha prayer and prayed with him, then you have caught one rak'ah, and you were required to complete three rak'ahs after his tasleem. If you intentionally prayed four rak'ahs, then your prayer is invalid and must be repeated. However, if the increase was due to forgetfulness, your prayer is valid, and it is sunnah for you to perform two prostrations of forgetfulness (sujood as-sahw) after the tasleem. Prayer becomes invalid due to an intentional increase of the same kind as the prayer, and one prostrates for forgetfulness. As for an increase due to ignorance, scholars have differed: some have invalidated the prayer because of it, while others do not consider it to be invalidating. The prostration of forgetfulness after the tasleem does not invalidate the prayer by its omission, nor is its time limit expired by the passage of time.