congrats!
please number questions if more than one
1. Not sure if actually printed yet but when it is the dates should all be the same.
2. Yes. Not sure if CBP will actually care but the airlines will before agreeing to board you, so best to be safe. It is only once you actually have a green card that passport expiry no longer matters for boarding (I still had an argument with an airline who wanted to refuse my daughter boarding on this basis despite her having a green card so…I know they watch for it). (Obviously if you are returning back home for a while you will get another passport for her while you are there anyway?)
3. I would not phrase it as you have the “right” to be out the US because that’s not how it works, but one year is the period before which you are not assumed to have abandoned the green card. Remember you will need to file taxes with the irs as a resident regardless where you are physically or earn income once you have activated the green card. And bear in mind longer than 6 months out will affect the time in which you can naturalize, if that is important to you.