You would not get a 7 day extension; you would receive a 7 day extension of stay denied stamp. This stamp begins on the day you receive the stamp, and is not added on to your existing "admitted to" stamp.
If you're an American, you will use that Affidavit from the US Embassy saying you live in Thailand, and you live at whatever address (it costs $50 USD). If you are not an American, ask your embassy for more information.
Try the Bangkok Bank in Centralworld (first floor). They only need a residency certificate issued by your embassy, and a 1,500 THB opening deposit (along with your passport)
So you didn't actually ask them what they expect for that letter? My opinion is to contact the consulate and ask them what should be on the contents of the letter