UK passports renewals are submitted at the Trendy in soi 13, although the passport is actually issued in Liverpool. A letter from your employer confirming your address in Thailand works, or a utility bill or rental contract in your name.
You (although technically it's her application) apply online and then submit everything at VFS in Bangkok. Now they scan everything to the UK where the decision is made. Settlement visas are no longer decided by the Embassy in Bangkok (although Visitor visas still are).
You will need to exactly meet the financial requirements as described by Gareth Youaintseenmeright Davies, and prove you are in a sustained relationship. She will need to pass the A1 English test (this can be done in Bangkok) and have a TB test. You will also need to show that she will have accommodation in the UK.
If she gets the visa she will be able to work in the UK. In 30 months she can complete the next stage on the way to Indefinite Leave to Remain and then, eventually, citizenship if she wants this.
Caleb is asking about Settlement visas not Visitor visas. There is no requirement to return to Thailand for a Settlement visa but there are tough financial requirements.
As long as you can show that you have enough money to pay for the whole trip (and you said you were only going for 2 weeks, so it shouldn't be too expensive) it should be fine. You have kids in school in Thailand, so include proof of this and any letters from the schools giving you permission to take them out of school for the trip to the UK. There is no requirement for your wife to be working, if she's a housewife!
You don't post it. Once the form is complete and you've paid, you set up an appointment in Bangkok (@ VFS) to hand it in, together with the supporting evidence, and for the biometrics. Then it's about a week before you get the email to say the passport can be collected, either with the visa inside or a notice of refusal.