Make sure you get your certificates legalised before you leave the UK as you'll need this for a work permit/extension later. Much easier to do it from the UK rather than from Thailand.
Not necessary if the new passport is issued in the UK, as it will be stamped upon arrival in Thailand. A letter is only required for a "virgin" passport.
All good advice. Your daughter doesn't need to be present when the application is submitted, in fact you can use an agent and not have to make the trip yourself, if you prefer. Importantly, you will need your long-form birth certificate to show that you are British Otherwise Than By Descent.