Thanks for that information, it helps me understand better what you want to do. Not that I have a lot of brilliant ideas, my situation is so different to yours that I doubt many of my ways would work for you. I still don't know how big that coop is, in feet, or how it is laid out inside. I tried to scale the length off the photo compared to the 12' run and I'm not sure it's as big as I came up with. I came up with an 8' long coop. That just doesn't feel right looking at those nests.
You are talking about five chicks already a month old. In Mt Juliet, if you wait until May to get them they should not need any heat. That eliminates one of my concerns. In Arkansas, I'd brood in the coop, taking chicks straight from the incubator to that brooder even when the outside temperatures were below freezing. You don't need to do that. Avoid it. That way, yours won't need brooding, just integrating. Integrating is where I see most of your issues anyway. That run is really small.
I don't know why you lost three of the four you tried brooding in the garage. If you feel you need to provide heat and you have electricity at the coop, you can do it out there, you will need a shelter for them anyway for integration. Or you could brood in the garage. If we knew what happened when you lost those three we may have suggestions on how to avoid that.
When integrating you might be able to section off a part of the coop, maybe using framing and chicken wire, to house the chicks. Or you could put a door in the far end of your run, fence off a section of your run (a 4' x 4' section would be enough), and put a shelter in there so they can sleep out of the rain. If your run is predator proof you can just use chicken wire, but you might want to lock them in the shelter at night if it is not.
I'd house them across wire fro at least a week, two might be better. Then I'd go with the safe haven/panic room concept. Out a hole in that wire big enough they can get through but small enough that the adults cannot. Then let them mingle as they will. after you are comfortable they will not be harmed by the two adults take he wire down. I don't use this concept but in your situation I think it might be a good idea.
If your coop is big enough you may be OK with seven chickens in there. It's not how much room is in the coop separate from how much is in the run, it's how much room total they have available. And the quality of that room. That's what Rosemary was talking about with clutter, do they have places they can hide behind. If you try this later in the year that outside area will be available every day all day. In the winter that may not be true. Your room is tight but not hopeless.
I don't bond with my chickens to the point I can just pick them up. Others do. If I were to try that I'd bring a chair and read a book around them. Bribe them with food to keep them close. Don't try ti pick them up from overhead (they think that's a hawk) but pick them up from the side. Be patient and let them come to you. It's not going to be a one or two day thing it will take time.
Good luck with it.