There is no formal process.
Something gets put on meta and if someone thinks it'd possibly be meaningful enough to be in the FAQ, it gets tagged with [faq-proposal] either at first or later on via normal Marquess of Queensbury editing rules.
Votes and/or discussion happen. When the mods are satisfied that it's now a [faq], we make it a [faq] by changing the tag.
The first question you cite, the headers question, has 51 upvotes and has been there for a couple weeks. Any meaningful discussion would have already happened; there's been no anti sentiment except a caveat of "be careful with h1's." That makes it clearly and overwhelmingly pass in our minds (51! That's just about unprecedented on Meta. I don't know why so many people care about formatting either, but eh.).
The PM one has been tagged faq-proposal as well, because people ask that question all the time. If you want to talk about it, yes indeed, do it on that question, in answers and comments and whatnot.
We don't make separate questions to declare the FAQ-worthiness of a question that is proposed to be a FAQ, that just adds confusion. "Here's a FAQ proposal!" "Let us have a proposal about the proposal!" This isn't HP or IBM, it's a community site... [faq-proposal] means "this is a FAQ proposal, please discuss its content and its FAQ-quiness." FAQquiness is not usually much of a debate, it's really a matter of "is this question asked a lot?" (The PM one sure is), as in, you know, is it a frequently asked question, or is it normative guidance we think everyone should read even if you didn't know to ask it, like something about headers or comment use or whatnot. There's not a tag for "official declarations!" other than [faq], so we use that for those too (I venture to say that's common in FAQs of the world).