REgarding outside food:
All food items must be contained in single serving bags within a soft-sided container or cooler that does not exceed 16" x 16" x 8". Metal, plastic or glass containers of any kind (except for factory-sealed, clear plastic water bottles, no larger than 1 liter, baby food, empty Nalgene bottles and juice boxes) are prohibited at Nationals Park. Only one bottle of water per person will be permitted
Does this mean you can bring food in a grocery bag, or does it have to be a soft sided cooler?
A grocery bag would be considered a soft-sided container, so you're fine.
