Although Thailand does not require 6 months validity as long as the passport is valid for the duration of stay, most airlines don't seem to accept this over an abundance of caution. I have been in this situation although over a land border and although the UK Embassy told me I needed 6 months, they were wrong and I was admitted. Your situation is tricky as it may be more difficult to convince the airline check-in. A letter from someone in authority in Thailand may help. Good luck.
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.