How are two missed rak'ahs from the Dhuhr or Asr prayer made up, while having only caught the last Tashahhud, and does one sit for the middle Tashahhud after making up the first rak'ah?

If you join the Imam in the third rak'ah, that is your first rak'ah. And when the Imam sits for the final tashahhud, that is your first tashahhud. When the Imam says the salutation of peace (tasleem), you stand up to perform the two remaining rak'ahs and then say the tasleem. You do not sit after the third rak'ah.