Almost certain that they have it on a regular basis. I was there on the weekend so I am not 100% sure

Classic French.

When I went, I noticed they had a nice outside sitting area. So I don't see why not. I believe a bringing the little dog for out door seating will be fine.

They have bread soups sandwiches quiches chicken salad pimento cheese coffees teas quite a delightful place.

No they do not sell bagels.

Check their website