would this hold two pairs of aviator style sunglasses?

Actually, they do. I just tried them with my husband's aviator ray-bans (not oversized) and they do fit. You need to make sure you put the top of the sunglasses in first (towards the deeper part.

My dog seems to like to chew my sunglass cases, is this strong enough to prevent that?

I love this case but it's not doggie proof. This flaps are magnetic closure so stay shut but easy access so if your dog wants to chew it he/she will find a way.

Would this fit in your average cup holder, like you would find in the car? How wide is it?

Yes, this will fit in a standard size car cup holder if you stand it on its end. I have a 2012 Ford Escape and it fits in all my cup holders. the size of the case is 6.25 inches across side to side (width), 3.0 inches high top to bottom (height), and 1 5/8 inches deep front to back.