How often do Americans use the stucture adjective + though + noun/pronoun + verb?

In speech? Functionally never. You may hear the variation with "as" once in a while. *(As) tired as I was, I still was able to drive home safely.* *(As) beautiful as Bermuda was, it was only our second-favorite stop on the cruise.* Will you see it in literature? Occasionally.

It’s technically correct but very outdated and poetic sounding. I’m sure people would understand but you might get some funny looks.

Rare though it is, I would understand that structure if I heard it.

I think people shorten it to make it not sound so literary (which is the best way to describe it I saw described). I would more often hear their second example but I'd hear it as "Even though it was hot as hell, I slept pretty well." Perhaps you'd hear someone use "as" for the first example, like this sounds more natural: "As hot as it was, I slept pretty well." Note I did change the sentence as if I was saying it about myself because that's what people tend to do in conversation unless they're talking about their baby or something.

Bloody though the crime scene was, she felt calm. Peaceful though the landscape was, they felt disturbed. Green though the intern was, she proved herself capable in that moment. Although the dog barked loudly, he did not wake. Although chest compressions were administered rapidly, he remained dying. Although the girl looked professional, she was denied service. The first 3 are definitely more literary and would be found in fiction, and the second 3 are more normal but still not casual. A more normal/casual version would be, “He didn’t wake up even though the dog was barking loudly.” “She felt calm despite the bloody crime scene.”

I would never in natural conversation. You may read this in poetry or books or other creative works, but you would probably never even once hear this in natural speech unless the person was doing a bit or something.

It sounds pompous.

It's very literary. You'll see it used in books, newspaper articles, documentaries, that sort of thing. But most people won't naturally use that expression in conversation.

If you say "with as hot as it was..." that's more natural for speech. The version from the picture though wouldn't really be said in spoken English.

“Well though” is what you’ll see more often

Most Americans do not know the order they will just know it sounds wrong.

not often, but honestly i structure sentences this way sometimes

i say a phrase like this maybe once a month.

Interesting though these comments are, the funny thing about English is people love the way English sounds in literature where these sorts of phrases are used, but they don't want anyone to speak that way. I speak thee sooth.

Do people actually do these mathematical equations on the fly while they are speaking?

Using "though" in that way sounds somewhat academic/stodgy, and I almost never hear it. Using "as" in its place is more common. Edit to add: If I had to guess, most uses of "though" in this way now are probably to try to sound more high class or fancy, or to keep a particular meter.

It’s not often. But I might say “As (adjective) as (pronoun) (verb)…” sometimes.

Weird as it may seem, you will encounter it fairly often if you read published books. It's a little more uncommon in speech, but you'll still encounter it.

It's very literary. It would not be out of place in a book, but I would never say it myself, pompous though I may be.

This exact sentence (hot though the night air was, and other variations with different word order) was in my uni admission exam, and it was the first time encountering it for me. This is the first proper explanation I see and it's still hard to grasp

Depends which century. 19th century? all the time. Now? almost never. Archaic as this rule is, you don't need to adopt it in your speech.

I doubt I've seen hot though the night air was. However, on the other hand, I have seen hot as the air was.

sounds a bit formal, but not unfamiliar. you'd be far more likely to encounter this structure in writing than in speech. oh... sorry... uh...... Though you may see it in writing, it is not commonly used in casual conversation.

With *though*: very, very rarely/never. With *as*: occasionally. *Hot as it was yesterday, he still refused to take off his jacket!* <-- It sounds fine, nobody would bat an eye at that. But it would sound even more natural with another "as" at the beginning.