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.
Correct. No fine or ban. They do however get an overstay stamp in their passport which could affect future extensions especially if they will be going to school in Thailand. Better to keep them legal if possible.
It's fairly straightforward if she has a solid job to return to (a "reason to return") and you can demonstrate that you have a genuine relationship. You can complete the online form for her but she will have to submit it and get her biometrics taken at VFS in Bangkok (or Chiang Mai). Allow plenty if time as there are delays in processing UK visas. If you want an agent, I recommend