The UK Embassy in Thailand do not legalise documents - it is done by the Legalisation Office in Milton Keynes - but they do certify passports. Appointments are made online.
Yes it's true. However if your kids plan to stay in Thailand long term and be educated here, they should stay legal and get proper visas and extensions.
Not difficult at all. You can complete the application online on her behalf but she will need to attend the VFS office (in Bangkok or Chiang Mai) to get her biometrics taken. The key factor is showing that she has a genuine reason to return to Thailand after her visit - like a job. She does not need to have money. If you are sponsoring her visit, you will need to show that you have enough money to cover her trip and somewhere for her to stay.