You did very well getting a UK passport for an 8 week old baby - some UK passports are taking ages to be issued. Presumably the baby was born in the UK? The best thing would be to get her Thai passport before taking her to Thailand, if you are all in the UK now.
No need to do anything for a while as she was born here, but once she leaves she will need to return visa exempt if entitled, or get her an O visa as your dependent and then she can be added to your Extension.
Actually Duncan is right. An IDP is just a translation of your national driving licence. You need a Thai licence if you're driving in Thailand for more than a few months.
Get them O visas as your dependents then they can be added to your annual Extension of Stay, assuming that's what you plan to get. They can continue to extend like this every year. No need for them to have their own Ed visa/extensions.
There is no cost if you do it yourself, apart from the cost of the new passport. Different agents have different fees - anything from 5-10,000 baht. Payment is processed in the UK so not sure if you can use Thai cards.
It's easy to do it yourself, if you're close to either Bangkok or Chiang Mai. The application has to be submitted in person at the HMPO office at VFS in Bangkok or Chiang Mai. You have to provide a colour photocopy of every page of the expiring passport, but you keep the original until the new one arrives. You'll also need proof of your address in Thailand (in English). Payment is by credit card. Currrently you might have to wait up to 12 weeks for the new one, which also has to be collected in person at VFS.