Is it permissible to pray at home due to the difficulty of walking to the mosque and rising from prostration and sitting from it due to thigh and knee pain, and will it be counted as prayer in the mosque? Also, what is the solution for the difficulty of washing the feet during ablution for the same reasons?

If going to the mosque causes you hardship and pain in your knees, then you are excused from attending congregational prayers. The reward for congregational prayer will be recorded for you as long as you are unable to perform it due to illness. As for washing the feet during ablution, as long as you are able to wash them, then it is obligatory to wash them, and it suffices to pour water over them without rubbing.