Welcome to BYC!
For your first question, regarding coop training, the answer is you will probably find it necessary to teach them - but they *may* take to it well enough on their own. I've had both experiences, but the majority tend toward the former. The exact approach will depend on your setup. If your run is such that you are able to enter and move about inside it, you can go into the run and "herd" them into the coop at dusk and/or pick them up and place them inside the coop. Another option is to open the coop (or go into it if it is large enough) and call the birds into you (using treats and a "rattle can" can be a great way to get them trained to respond to the noise by coming running).
On the second issue - I prefer to keep the boxes closed off to prevent them from developing a bad habit of nest sleeping/being in the nests before they are ready to avoid waste fouling the nests. This is more of a challenge if/when you add the next generation to your flock and have actively laying birds in with younger birds as closing the nests is not an option. In that case you would simply remove the younger birds from the nests if/when you catch them attempting to sleep in the boxes, etc.