I keep straw in the nest, and hemp bedding in the coop. They haven't had any problems with the bedding. For a chick safe waterer, put some rocks in a low bowl, preferably one that won't tip too easily. The idea is that they have to drink between the rocks, but if they fall asleep in the bowl...
If they're at all up to the task, they should just be hissing and pecking at you if you stick your hand under them. I wouldn't be too worried about taking a peek.
I've moved eggs without issues, but it's always a risk.
Compared to the whole incubator and brooder business, I must say the broody method is so much less work. I'm now on my second batch of broody hatched chicks, for the first one we separated the chicks and broodies from the main coop, but...