You have just over 4 square feet and three chicks. I've kept mine in the brooder until they are fully feathered out (usually around 4 weeks, never more than 5 weeks). I had 27 chicks in a 3' x 5' brooder or 15 square feet, which gave them just over 1/2 square foot each. It is a different situation because of the difference in the numbers but you could probably keep them in that brooder until they are feathered out, especially if you are worried about the fire danger having a heat lamp in the coop. You will need a top on the brooder to keep them from gettting out. It is amazing how fast they learn to fly up that high.
If you do put them in the coop with a heat lamp, make sure the heat lamp is very well secured. Do not depend on the clamp that comes with it but support it with wire or something so it cannot fall. They will bump it and knock it off.
With all that said, my brooder is on the coop from day one. I keep one area the recommended temperature range and let the rest cool down as much as it wants to. With my last batch, the area under the lamp was in the low 90's but the far corner was in the 70's. Within 2 days , they were playing all over the brooder, including the cool corner. They would go back to the heat when they wanted to warm up. I think raising them in the coop helps in the integration into the flock.
When you can mix them depends on a lot of things, flock make-up (age, numbers, and sex), how much room you have, whether you are around during the day to watch them, some of your management techniques, personalities of your chickens, and other factors. What I recommend is to house them where they can see each other without getting to each other for a while, let them mix where they have a lot of room and you can watch for a while to see how it goes, have separate feeding and watering spots so the older ones don't keep the young ones away from the food and water, and just move at your own pace. Don't rush it but don't be afraid to take a few chances. There will probably be some squabbles but if the young ones can get away, they will normally be OK. I always kept mine where they could see each other, let them free range over a big area at about 10-1/2 weeks old, and by 12 weeks old they were sleeping on the roosts with the big ones. They were crowded on the far end of the roosts as far from the big ones as they could get, but they were still up there. And I could leave them locked in the coop together for a couple of hours after they woke up without them hurting each other.
All our circumstances are different. No one answer is right for all of us. Hopefully you can read what others did and pick and choose the parts that seem to fit your situation and come up with a plan that works for you. Good luck!