I concur with most of the above. Under 2 (if you don't pay the fare and have a "lap" child), you may or may not need proof of age. I've done it many times with my kids and have never been questioned. I've read of others who have.
Leaving the country, yes you will need a passport for your children.
Under 18, I'm not sure of. I would think that if they are old enough for a drivers license then they would be "carded", but that's personal opinion. I've never checked into it so 18 may very well be correct.