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Alot would depend on what the temps. are like where you live, but I can't see the sense in waiting until the chicks are 12 weeks old before allowing them access to the whole coop and/or the run.
Here's the temps. you will need to maintain in at least a portion of your coop for the first five weeks - 90 to 95 degrees the first week (this varies because I've never had a chick that liked it that hot), decreasing by 5 degrees a week until such time as you are down to 70* (end of week five) at which point the chicks will normally not require supplementary heat.
The chicks don't need their whole coop to be at those temps.; in fact it makes good sense to give them an area a bit cooler so they can learn to self-regulate; going under the heat when they need a quick warm up. This is the only reason it makes sense not to give them full run of the coop at first. You need to keep them close to the heat source until they've learned to self regulate.
They can also go outside on nice, windless days when temps. are above 70 or so. Be prepared to have to shoo them back in the coop though, since they won't have a broody mama to teach them. Keep in mind that small chicks are most vulnerable to predators, so that run needs to be super secure.
Also, and this is extremely important, be sure your heat lamp is secure in at least two ways! An example would be hanging from a stout chain and zip tied too. More than one coop fire has been started by a heat lamp that fell onto the bedding.
Now that I have raised brooder chicks and watched my broody hens raise chicks, I have to say that I like the broody's method better. She doesn't keep her chicks under her full-time, she lets them roam a bit, always on stand by to call them back under her should they start to get chilled. I've seen chicks out when there was still snow on the ground and they grew up fine with a mama hen there to take care of them.
The ideal situation would be to mimic the care of a mother hen as much as possible.
Also, don't worry so much about the bedding getting soiled. Keep it clean but not sterile. Chicks need to be exposed to germs to develop strong immune systems.