I got 4 day old chicks almost 5 weeks ago and the temps are dropping. They are in a bought coop from tsc by themselves. I am just concerned they are going to be cold.
They won't be any colder than they can handle. Being 5 weeks old and kept outside, their down and feather coats will keep them warm enough. Your job is not to give them a warm place, it's to give them a place they can keep themselves warm. You do that by giving them a dry place out of direct wind.
Your biggest risk once it droops below freezing is frostbite. The biggest cause of frostbite is lack of ventilation as that allows moisture to build up in the coop from their breathing, their wet poop, waterers, or spilled water. Those
TSC coops (and practically all prefab coops) are notorious for not having enough ventilation. Could you post a link to the one you got, maybe we could give you suggestions on how to do that without creating breezes hitting them directly.
I was wondering if it would be wise to try to integrate the chickies into the rest of the flock in the shed with everyone else or do you think they will be ok in their coop.
With proper ventilation they will be OK in their coop. But I think it is still a good idea to start the integration process. You want those pullets in that coop when they start laying so they can use the nests. Aart loves to complicate things, I sometimes tease her about that, but her method is very safe. Sometimes these integrations are as smooth as you saw this spring, sometimes they can be disasters. You can't tell ahead of time. I strongly believe the more room you give them the better off you are.
My basic suggestions are to house them where they can see each other for a while but not touch. Have separate and widely spaced feeding and watering stations so they can all eat without causing conflict (sometimes the older ones bully them away from the food or water). Give them as much room as you can and improve the quality of that room by giving them places to hide under, behind, or over by breaking the line of sight. Take your time, don't rush it. I like to let them range together during the day before I try to put the in the same coop to sleep. Let them work it out on their schedule as much as you can. I've never used that safe haven/panic room but in many cases it is a great idea.
I don't know how your two coops are arranged. If you can, I'd position the
TSC coop next to the other one so they can see each other. Make a connection between the runs. After at least a week and when you can observe, let them roam together during the day. Let them return to their own coop at night. After they have roamed together during the day for a month try to move them to the main coop at night.
I consider this overkill, Aart can tease me back if she wishes. I don't do it this way. My brooder is in the coop so they are essentially raised with the flock. I have a lot of room, at least an order of magnitude greater than the recommendations you see on here for outside. My large main coop is cluttered, they have a lot of places they can hide under, behind, or above. I have weather they can pretty much be outside all day every day. Your weather may create some problems for you. At five weeks I just open the brooder door and let them mingle. I think you are in a different place so I would be more cautious.
Just do the best you can and observe. It will probably work out with less drama than many people would expect.. And don't worry about cold, worry about ventilation.