I know nothing of "weekly pay". I don't get paid weekly and haven't heard of any other teachers who have.
I receive pay monthly of 1/12th of my salary. The district did not ask me how I would prefer to be paid. All salaried employees are paid on the same schedule.
There was an option during my first year with the district to receive a 13th check and have the salary split into 13 equal portions so that I would have pay for the month of August instead of working for nearly two months before receiving any compensation. I utilized this option during my first year (as there was no other viable option).
I had not heard of teachers collecting unemployment during the summer months.