My question is regarding transferring them to their new coop. I've read that you are supposed to lock them in the coop so they can establish it as their homebase. But the times recommended vary widely from 24 hours to weeks (seems extreme).
You can get extreme answers to any question on here. Tp some people what I consider extreme may be absolutely necessary. So take any opinion as an opinion. Somebody else will have a different opinion.
To me there are two parts to this, depending in your conditions. If you turn them loose can they wander off and never return? Or do you have fences to contain them? If they can disappear over the horizon you probably want to contain them until they accept your place as home. A story about this. Dad free ranged his chickens on a farm out in the country. Most slept in the hen house but a few slept in trees. When Dad got new chickens he'd take them to the hen house area and turn them loose. They were usually around 4 to 5 week sold. They'd stick in the area because it already had a flock. Some would eventually learn to sleep in the hen house but some would sleep in trees. They did not just wander off. Eventually he got to use their eggs and eat the boys for meat.
The other part of this is that they often need to be taught where to sleep. Not everybody is happy with them sleeping in trees or in the run. If you lock them in the coop section only for a few days they often learn to sleep there and will go in at night on their own. Often does not always mean always. I'd lock mine in the coop section only of my grow-out pen for a week and they would still try to sleep in the run when I first let them out. They had to be taught to go in at night. I think that had to do with the configuration of my coop and pop door. Mine preferred to sleep under the elevated pop door instead of going up through the pop door and inside to sleep. That coop had a lot of hardware cloth so it was plenty light.
If you just turn them loose in the run they are unlikely to go to sleep in the coop on their own without being taught. You are practically guaranteed that you will have to train them.
Many people are very successful on getting them to put themselves to bed in there after housing them in the coop only for a while. Others have less success.
My last question is when can I let them out in my fence enclosed backyard?
I don't know where you are located or what your weather is like, but at 6 weeks they should be able to handle practically any weather unless you are somewhere really extreme. From a weather standpoint, they are ready.
The other part is how are you going to put them to bed. You may have to train them.