I’ll answer my own question. Yes! They have the stamp in the Ranger station near the entrance to the Filene Center. No problem!

"Patrons may bring water into the covered seating area. Other beverages (wine, beer, mixed and soft drinks) may only be brought into the covered seating area in official Wolf Trap Cups, with the lid in place. Wolf Trap Cups are reusable and spill-safe, and may be purchased at the Gift Shops and Concessions."

Regular theater seats (It sounds like you are not talking about lawn seats?)

Nothing really. The shows combine with good weather are the attractions.

Convenient, but expensive. If you don't have time to cook, you can always put together a picnic from one of many grocery stores in the area. They all have a good assortment of ready made foods.

We have not had any issue with mosquitoes, but perhaps take repellant, just in case. It is wonderful there, that noone judges your attire. You can go casual as you like or dress up. You see a little of both. I do suggest to take a light jacket or wrap, as it can get a little cool, as the evening progresses. Have fun!

Very low beach chairs. But best to just bring a nice fluffy blanket.