That's talking about a Thai marriage certificate - the OP is asking about a US marriage certificate. The US Embassy can authenticate the signatures of certain MFA officials, because they are given copies of the names and signatures of those officials to keep on file. The embassy has no way to authenticate the signatures of state officials in the US.
They're using the wrong terminology - what they really want is an authenticated copy, not a notarized copy. To fully authenticate the document, you would need to have the seal and signature shown in the photo authenticated by a high level office of the State of Oregon, probably the Secretary of State's office. That officer's signature would be authenticated by the US Dept of State in DC, and THAT signature would then be authenticated by the Thai Embassy in DC. It's a very cumbersome and time consuming process that is total overkill for something as minor as a visa extension, but they can legitimately request it. (There's an international convention that greatly simplifies the authentication process, but Thailand isn't a signatory.)
Other people have said that their embassy notarized their marriage certificate. If the embassy will let you do a self-serving affidavit stating that the certificate is genuine, the MFA in Bangkok could authenticate the signature from the embassy and immigration would probably be happy. As a legal procedure that's totally meaningless, but all they really care about is the form, not the substance.
A notary seal can be raised or just a rubber stamp, depending on the purpose for which it's being used. (Someone who plans to fax or scan a document often prefers a rubber seal.) I was a notary for my entire career and had both types of seals.
"After December 31, 2015, the Department of State will no longer add extra visa pages to U.S. passports.
"U.S. citizens applying for passports outside of the United States will receive 52-page passport books. The 52-page passport book will cost the same as 28-page passport book would."
There's never been an extra fee for the bigger book, but automatically issuing it to Americans abroad is something they started at the beginning of 2016 when they stopped offering extra page inserts.
Just to clarify, there is no additional fee for the larger US passport book (and it is now issued automatically if you're applying overseas, but applicants in the US can get one by ticking the appropriate box on the form).
You are not on a multiple entry non-O like the OP, you're on a retirement extension and have a multiple entry reentry permit. Unlike the OP, you don't have to leave the country unless you want to. There's no fee for coming back in with a reentry permit - the OP just paid for transportation and facilitation (and presumably an entry fee for Burma) in order to get his new three month stay as quickly as possible. If you take a trip somewhere and reenter normally, Thai immigration will just stamp you in with no charge.
You're generally correct, except that deed polls aren't used in the US - they mainly exist in Commonwealth countries. But yes, the US will be fine with her using any name that reasonably describes her and that she uses habitually.
For passport purposes, the US does not require a separate legal name change procedure for applicants who get married or divorced. The marriage or divorce certificate on its own can be used as proof, and an applicant can use either the current or previous name as she wishes, if that's the one she uses in everyday life.