I spend a lot of time in Door County. Most of what was mentioned above is quality information.
Favorite breakfast spotsVillage Cafe - Egg Harbor
Scaturo's cafe and bakery - Sturgeon Bay
Rusty Tractor - Carlsville
Upscale dinnerLure - Sister Bay
Chop (already mentioned)
Casual Dinner/lunchCasey's Smokehouse - Egg Harbor
Husby's - Sister Bay
Sonny's Pizza - Sturgeon Bay
Island Fever rum Bar & Grill
Cool places and things to doSister Bay Bowl - has to be one of the world's oldest bowling alley with manual scoring.
Washington island (take the ferry). Last ferry back to the mainland is at about 5:30. Be in line at least 45 minutes before that, or you are stuck for the night.
Cana Island lighthouse
Kayaking in Baily's Harbor - you can see down to some old shipwrecks in crystal clear water
Frank Murphy park in Egg Harbor - nice sand beach and great swimming
Book a salmon fishing charter in Sturgeon Bay
Plus lots of wineries, breweries, distilleries
IM me if you need some more details.
Edit to add: if you go to Washington Island, make certain that you visit Schoolhouse beach on the north shore. It is one of the wonders of the world. IIRC, there are only two or three places in the world with the large smooth stones like they have there. You can swim if you want, but expect the water to be colder than the beer at Twin Peaks.