The right sentence, 선생님께서(는) 학교에 가십니다.
선생님께서는, 선생님께서 both are right, just depend on situations.
There's some difference between two.
When you want to emphasize the subject, you can use (는),
but when you just refer to the fact or situation that the teather goes to school,
it's better to use without (는).
The most reasonable sentence, 어머니께서 진지를 드시네요. (잡수시네요, possible)
But, the object 밥 dosen't go with the usage of the verb.
밥 goes with the verb, 먹다 which is used between friends or
when older people talk to younger people.
어머니께서는 밥을 잡수시네요.
(actually, the sentence is sort of wrong, but to explain something, I will let it be)
The sentence means that she's eating, but the others.
We could expect some situations through the sentence, for instances,
only the mother is eating rice, but the others? not eating or eating another food.
Hope to make you clear.