I am having a similar issue, my 2 girls (my first chickens) have decided to lay their eggs under the screen ladder that exits their coop into the fenced in run. I have now removed the ladder to see where they try next.... if they don't go for the nest boxes inside the coop, I'll try locking...
Here is our plastic storage bin style brooder - so far so good for 4 baby chicks- now 5 days, I have the brooder on a table in the garage, garage temp stays about 60 degrees, with heat lamps hanging securely by chain from the garage door motor keeping the brooder about 95 in the warmest area...