Is it okay to have pets?

I believe you can bring pets, but there is a limit on how many. There are also certain requirements that must be met in order for the pet to enter Korea.

I would suggest checking with the school that's employing you, especially since they are also securing the accommodations for their teachers (hotels, apartments, etc.). Also, check out Korea4Expats.com because they have an article on this topic dated from September 2017, so it's fairly recent. The local Korean embassy would also be a good resource too.

I think that would be a case by case basis. Most often schools provide apartments rather than the teacher arranging that themselves.

In Korea the deposit to rent a place is insanely expensive (like one years rent). So for teachers to arrange their own accommodation is difficult.

Often times teachers are also placed in small one room studios know as officetels. The teacher wanting to bring a pet should also consider the small space they will be living in.

It depends on the apartment you end up in, I have many friends with small cats and dogs so generally yes, but not all apartments will allow it.