This is an interesting thread. I used to stealth camp when I was so poor that the price of a campsite in a state park was a budget-buster (and there were no such things as hiker/biker sites.) But I usually did it in a park - in the woods or in the picnic section after dark. I had a few uncomfortable incidents and gave it up quickly. However, lately I've felt tempted to avoid campgrounds due to the nature of the people one often finds there - people who drink and play their stereos loud until late at night, throw frisbees into your campsite (tent) and run to get them without acknowledging your presence, that sort of thing. I'd prefer a grassy spot in a KOA to a regular campsite in some campgrounds I've been in. For that reason, if I'm in a national forest, and there are no hiker/biker sites around, I'd prefer camping out in the woods, alone. I mention hiker/biker sites because, so far, I've never had a bad experience in one. Bikers (like me) tend to be pooped in the evening. They're quiet and they go to bed early!
Touring with kids is definitely do-able. I recommend reading "Hey Mom, Can I Ride My Bike Across America?" by John Seigel Boettner. He took some middle-schoolers across the country. He's done it again since, but this time he produced a DVD of the trip, instead of writing a book. I got a copy from ACA.
I've toured with kids. You have to find kids who are into it, and most wouldn't be, but there are a few. Then you have to spend some time teaching them, most importantly safe riding practices and proper shifting, cadence, etc. so they can cycle efficiently and not tire themselves out too quickly. (I was riding with a 12 year old last weekend, riding a 21-speed bike, who said, "I always ride in [gear number] seven. It's my favorite. Seven is the best! She was exhausted at the end of a 22 mile ride. Gee, I wonder why? I tried to give her some tips on shifting, but she didn't seem to want to hear it. After all, seven was her favorite.)
I took a short tour with my son when he was 8. Back then he was tough (now that he can drive he won't get on his bike.) I rode behind him the whole way so I could watch what he was doing and shout instructions if necessary. Your 14 and 12 year olds probably won't need that, but kids that age often get cocky, or lose focus. They have to be ingrained with the idea that it only takes one moment of carelessness to end up maimed, brain-damaged, or dead.