To secure a door cut into a cardboard box, I use popsicle stick with a screw in the middle, then screwed into the side of the door flap. Swivel the sticks to "closed" positions to hold the flap closed.
No, you do not want to bring the heat source outside with the chicks. By the time chicks are two weeks old, they are going long stretches on nice days without the need to warm up. The main reason to take chicks outdoors is to harden them to cooler temps.
By age three weeks, chicks no longer need a heat source during the day when the temp is in the 70s.
What about at night? My youngest are 3.5 weeks old and they are in the garage which gets to about 50 at night.