My read of the situation is the likelihood of getting a dead bird, mouse, snake, etc. is pretty low in hay. It’s not unheard of, but I personally have never gotten anything in my 3 years of constant purchasing. If you want to check, feel free, but I personally never do…. haha. Call it wishful thinking!
In terms of storage, I think the most important basics are 1) low to no sunlight and 2) dry. Hay will get yellow fast if left out in the sun and can get moldy/go bad if exposed to water and no able to dry properly.
I think whether you keep it in the bag/box it came in or a plastic container, either is fine. Rabbit owners tend to access the hay supply daily because you are cleaning litter boxes and topping off/replenishing hay all the time, so air flow will get there.
Personally, I get 10lb of hay and it is shipped in a cardboard box. I always have a “back-up” box that’s simply kept in the back of a clothing closet, so no sun light or exposure to the outside world, but nothing real specific with forcing airflow or anything.
The answers provided in this discussion are for general guideline purposes only. The information is not intended to diagnose or treat your pet. Seek the advice of your veterinarian or a qualified behaviorist.