Is my action correct in re-praying the 'Isha prayer with only two rak'ahs after the Farewell Tawaf, knowing that I doubted not having caught the first rak'ah with the Imam and a long time had passed since the prayer?

If you performed the opening takbir behind the Imam while he was rising from bowing (ruku'), then you missed the rak'ah, and you should have performed a substitute rak'ah after the Imam's salam. Since you did not do so and a long time passed, the prayer has become invalid, and you must repeat it in its entirety.

As for your praying two rak'ahs shortened (qasr) for 'Isha', it is not valid, because you are required to complete it as long as you prayed 'Isha' behind a resident (muqim), and you must repeat it in full.

Ibn 'Uthaymeen held the view that it is permissible to shorten the prayer after it has become completely invalid, but it is more prudent to complete it.