That's what I thought. Flying from BKK is much more convenient than flying from Laos, so some expats who fly regularly must need to cross the border a lot to do so. The METV takes care of that.
So you can do unlimited border crossings if you are getting a flight from BKK? I didn't know that. Presumably you have to show a flight ticket out of BKK?
The 2 visa exempt land entries per year must be having a huge impact on westerners who live in Cambodia and Laos and rely on trips across the border for decent/affordable shopping, healthcare and flights from BKK.
You can travel internally with just the (Thai) birth certificate. If you're British, getting the baby a UK passport (in Thailand) is easy and only takes a few weeks. There is no DNA test; just the baby's birth certificate and the British parent's birth certificates.
And, as another thread has discussed, there is a chance you may be asked to show it when re-entering Thailand, so it should certainly be carried when travelling.
You can convert the Tourist visa to an O visa at Immigration in Bangkok, as long as there are enough days left for it to be considered. I did this for my wife and then she was added to my Extension. If there is not enough time left, she'll have to get an O visa from a Consulate outside Thailand.