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.
I turned up to get one for my wife, without her, and they initially refused. However I was referred to the Supervisor and pleaded and grovelled, with lots of wai-ing, and she relented.
Kids are not fined or banned for overstaying but their passport is stamped with an overstay stamp. If you're a tourist that should be of no concern. If you plan on staying here for a long time or if the kids will go to school here, I would keep them legal.
You have to include colour photocopies of every page of your old passport, including blank pages, when applying for a new UK passport from Bangkok. I don't think they will issue a new passport if you have lots of unused pages.