I've flown alot and have NEVER been asked for I.D. or birth certificate ever.
I've flown with a 2year old on my lap and I was miserable so never wanted to cheat and try it again. I'm not against cheating I'm a single parent who struggles financially and I cheat all the time at like restaurant buffets. I know I'm bad, oh well. My daughter is a small 6 year old and I can pass her for 4 all the time. She's good at pretending to be 4 too.
When my daughter was 3 it was totally worth the money to buy her a seat, airplanes are so crammed and tight it's really hard to hold a toddler on your lap for hours.
If they mention anything about your child's age like act suspicous say "gosh we get that alot, and she's only 22mos. old"
My ex-boyfriends father took my daughter to the zoo last summer. He lied and said my almost 5 year old at the time was 2 to get her in for free. She was in a stroller,so I guess that convinced them, and I would NEVER go that far but noone said anything. I couldn't believe he did that, that's really pushing it.