The only thing you have to worry about is if immigration want to visit the house when under consideration for the extension. They will want to see you in and outside the house and take pictures of the sign showing the address with you next to it
You will have to get your stamp transferred from wherever you got your last visa or extension. You MAY be extensively interviewed at the border/airport on entry enquiring why you are not in class. They may follow up with visits to your place of education and interview the staff.
As the others have stated you get a valid stamp only until the end of your passport. You then have to re apply again following the same criteria to get a 1yr extension in your new passport.
You both have to attend the Royal Thai Embassy in London , best guess is you fill in the application forms there and provide ALL documents at the same time