The characters only meet through their letters with a couple of exceptions, so some say the love story is unbelievable. For me, it reflects the extreme isolation and loneliness of their work and how even minimal tenuous companionship of a peer would satisfy a gaping need.
This is a perfect summary of that point.
She didn’t expect him to pay $1300. She asked what he can contribute and he provided a number. Also it wasn’t 1300 for rent, it was 1300 for everything according to him, rent, food, car, etc.