1. It's a very pretty area for biking, with nice, small towns, beautiful scenery and flat topography. It can be very windy, though, so take that into consideration when planning your trip. Depending on the wind, it can offset any advantage from the lack of hills. If going in the spring and/or summer, be prepared for lack of shade and bugs (depending where you are).
2. Roads are often either large and busy (e.g., rt. 13, 113, 50, 301) or small and without shoulders. While the small roads are often low-traffic, they often have high speed limits (50 mph) and little/no shoulders, with a drainage ditch on each side (this is pretty typical for rural roads throughout the peninsula and even down in Virginia). Although this sounds scary, I've never had a problem with cars on the small roads, even when riding with my family (on a tandem + trailer, for example). I'm not sure I'd feel comfortable with a young child (~7 y.o.) riding his/her own bike on them, though. If on a tandem or trail-a-bike, no problem.
3. Official camping can be somewhat limited. Over on the sea coast you'll find a number of campgrounds, most of which cater mainly to the RV crowd. Camping on Assateaque Island (National Park and state park) is really nice, but both the NP and state campgrounds fill up fast in the nice seasons. I wouldn't count on rolling up and getting a site on the spot. Best to reserve in advance. In Assateaque I greatly prefer the bay-side sites over the oceanfront sites.
4. With the exception of the Chesapeake Bay Bridge-Tunnel, I believe most bridges in the area are fine for biking. Granted, they may have narrow/non-existent shoulders, but you can ride them. It's fine to bike into Chincoteague, to answer that question. (Personally, Chincoteaque doesn't do a whole lot for me and I wouldn't ride that far out of my way to see it, but that's just me. It's very touristy.) The Chincoteaque refuge itself is really nice and worth seeing, but the town isn't, IMO, given the detour needed to get there.
5. As you get farther down the peninsula (in Virginia, for example) you'll find that many of the towns become smaller and, arguably, less well-off. Don't count on finding lots of/any services in towns (food stores, etc.). Many of the services are clustered out on the main highways, like rt. 13.
6. Rt. 13 is the main N-S route and is multi-lane and high-speed. It is bikeable, and has shoulders, but isn't always the nicest route. Side roads are not terribly direct, but are pretty. Rt. 113 is smaller and also a N-S route, and is more bikeable. But, it's very busy in summer beach season.
7. A really pretty area not to miss is Blackwater National Wildlife Refuge near Cambridge, Maryland. Lots of really nice, lightly traveled roads, most with water views of some sort. It's really nice. A good family tour might take in that area, along with Easton and St. Michaels, for example. This would be my suggestion for a first tour with kids there. In fact, we're thinking of doing it this summer with our 5-year-old son on our triplet.
8. The Chesapeake Bay Bridge-Tunnel does theoretically offer rides across with advance notice. You might be better off hitching a ride across with someone who has a pickup or van or whatever. If going N-S you might be able to easily do this as there is a rest area right before the bridge starts. Going S-N there isn't as much of an obvious place where people stop, as the access road is more-or-less a highway in Virginia Beach.
9. You should check out this trip report on crazyguyonabike. A family who did a tour on the peninsula: http://www.crazyguyonabike.com/doc/?o=RrzKj&doc_id=3293&v=8x