How do I complete the Asr prayer after catching the third rak'ah, and do I sit for the Tashahhud in this rak'ah, or do I continue for two more rak'ahs and then sit for the Taslim?

If you join the Imam in the third rak'ah of 'Asr prayer, remain with him until he says the tasleem, then make up the two rak'ahs you missed. You will only sit for the final tashahhud, because you already sat for the first tashahhud in what was your second rak'ah. The preponderant opinion is that what the worshipper catches with his Imam is considered the beginning of his prayer. If the latecomer joins the Imam in the last rak'ah, he should sit for the first tashahhud in the first rak'ah he makes up. As for reciting Al-Fatihah in the missed rak'ahs, the majority of jurists recommend making it up, while others hold the view that the recitation does not need to be made up.