A trick I use for minimum aquarium size is that the two different sides of the aquarium should add up to more than the length of the snake. Brown snakes are pretty small so a 5gal may be fine.

Shedding frequency involves my factors including; temperature, humidity, growth, and season. An adult snake may shed only once a year, but a young snake could shed every month or so.

Even the happiest and most contented of snakes will try to escape at somepoint - They aren't generally very rational creatures. That the snake tried to escape should not be a reason to question it's welfare.

Good luck with your snake! I hope she will be just fine.