What should I do in the middle sitting after catching the second rak'ah of the Asr prayer, subsequent to rising from bowing and prostrating with the congregation? Should I recite the glorifications and the tashahhud, or should I remain silent and begin recitation and glorification in the following two rak'ahs?

If you enter the mosque and the Imam is rising from ruku' (bowing) or is in sujud (prostration), you must say the takbirat al-ihram (opening takbir) and follow the Imam in his current posture. The rak'ah (unit of prayer) in which you did not catch the ruku' is not counted. It is prescribed for you to recite the dhikr (remembrance) legislated for that posture, and it is not permissible to remain silent, for the Prophet, peace be upon him, said: "When one of you comes to the prayer and the Imam is in a certain posture, let him do as the Imam does." Even if you catch the Imam in the last tashahhud, say the takbirat al-ihram, sit and recite the tashahhud with him, then stand up and make up what you missed after the Imam’s taslim (salutation of peace). As for remaining silent until you stand for the second rak'ah, that is not prescribed, because silence in prayer is forbidden.