Is it obligatory to go to the mosque for other congregational prayers like 'Isha and Fajr, knowing that there is a mosque ten minutes away by car, and that other prayers are performed congregationally with the wife at home?

Congregational prayer in the mosque is not obligatory except for one who hears the call without a loudspeaker. The distance mentioned is far, so it is unlikely that the call would be heard from it. Therefore, congregational prayer in that mosque is not obligatory for you. However, if you do pray there, your reward will be greater. If you pray at home with your family, there is no harm, and you will receive the reward of congregational prayer. As for Friday prayer, it is obligatory for the people of the city to strive to attend it, even if they do not hear the call.