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
I'm in a similar position (moving to Rayong but still employed by my Bangkok school). With the growth of online teaching, could there be an issue if your home address (where you teach online) is a long way from the place of employment address shown on the work permit? Maybe it's very common these days.