As US-citizen you will be visa exempted for a stay of up to 30 days. The stay can be extended at any immigration office by another 30 days for 1900 Baht.
I also had to provide that at the consulate in Switzerland. They just looked at the letter and saw that there was a logo and some business address. That was it. They didn't even read it, they just wanted it for the folder.
Is the visa valid until 4th November or your current stamp?
If it's the visa you can re-enter Thailand before the 4th November and you will get another stamp for 60 days. That stay can be extended by another 30 days at immigration for 1900 Baht. So you could be good until end of January / beginning of February.
Jason already mentioned, where you can get a new METV.
Yes, immigration has a database. Changing the passwort helps at the consulate / embassy, because they only go through the pages in the current passport to decide if they give a visa or not, but it won't help when entering the country.
Hey, you are right, the entries with a visa do NOT count when it comes to the entry limit with visa exemption. So, yes, you can do that border run and you should be fine. Make sure to have all 3 proofs (10'000 Baht, proof of onward travel and proof of accommodation) ready, just in case they decide to check.