I usually take them out the day after they hatch, but only for a minute. Be very gentle with them, they are quite fragile at that age.
If the parents are scared of you it might stress them a little, but no more than if you walked by and they flew off. Handling a baby won't make the parents...
Sounds like you have researched a lot and have great plans!
Well, if you handle their children you could get some cuddlers! And eventually I'm sure these ones will get used to you.
Okay, if you spend lots of time in there with them they will tame quickly! They must be eating then! As someone else said though it's best to give them regulated meals. When they are older they will probably use the perches more.
Congratulations on being a pigeon owner! They are beautiful and look healthy! And it looks like you made them a nice place to live. I hope they turn out to be great breeders and racers!