I have flown both with and without the carseat for my daughter. I thought it was a pain in the butt to carry one more thing thru the airport and plane and she was way happier to sit on the seat at 2.
No ID is required, last month I flew with both kids and my son was on my lap, he was not even listed as being on the plane which was really strange to me. My daughter had her own ticket but was not required to have ID.
I would recommend a DVD player IPod or video player in addition to few toys and books and lots of snacks. I also loved earplanes and made sure I brought a juice to sip to help with ear pressure.
Good luck! Be prepared to give your kid 100% of your attention and not to be able to relax and you will be ok.