Yes. UK HMPO is in the trendy building. More info here

So the proof of address doesn't need to be a Thai address?

I have already changed my address with the UK HMRC to my address in Thailand. I used a recent reminder letter from them about my annual tax self assessment which was obviously in English, so no translation required.

Yes. UK passport.

I recently exited after obtaining a new passport. I showed both passports at the airport immigration. I was taken to a separate desk where a lady placed a stamp in my new passport with dates from my old visa extension and a comment stating that my old passport was full.

The point is, I think you can use both passports to travel with a valid visa or extension stamp in the old passport.

You only need to transfer an extension stamp if you are planning to extend again or get a proof of address etc.