Is it permissible for me, as a teacher whose village is 200 km away from Mecca, to pray Dhuhr and Asr shortened and combined upon arriving at my home in Mecca, knowing that the time for Dhuhr prayer begins while we are on the way, and my arrival home is after Asr?

You are not permitted to shorten or combine prayers in your home after arriving from travel, unless the time for 'Asr prayer began before you reached Mecca. In that case, you may combine the Dhuhr and 'Asr prayers by delaying Dhuhr (Jam' Takhīr), provided you complete them (i.e., pray them in full) if you pray them after entering Mecca.