"Despite the fact that x" <-- phrase, a clause, or phrase + a clause?

Question

I know that “despite” is a preposition. However, I am confused about its role in the following sentence (and similar sentences that use “despite the fact that”):

SENTENCE: Despite the fact that I was tired, I stayed up late.

1. Is “Despite the fact” a prepositional phrase and “that I was tired” an adjective clause (that describes the fact)?
2. If #1 is true, can "despite the fact that I was tired" add up to or be simplified as 1 chunk that gets its own name? Like is it 1 long phrase since the clause is modifying the phrase? Or is it 1 whole dependent clause because it is subordinate to the main clause?
   This textbook (image pasted below) seems to say that it is categorized as 1 big dependent (aka subordinate) clause? Is this true?
3. Finally, if “despite the fact that I was tired” IS a subordinating clause, does that mean that “despite the fact” acts as a subordinating conjunction (similar to although), even though “despite” by itself is a preposition?

Answer 1

Is “Despite the fact” a prepositional phrase and “that I was tired” an adjective clause (that describes the fact)?

You could call "that I was tired" an adjective (or relative) clause, a nominal clause functioning as an appositive, a complement clause, etc. There are many terms that people use, but in any case it provides further information about "the fact". Thus, "the fact that I was tired" constitutes a single phrase. You couldn't construe "despite the fact" as a phrase while leaving out "that I was tired".

If #1 is true, can "despite the fact that I was tired" add up to or be simplified as 1 chunk that gets its own name? Like is it 1 long phrase since the clause is modifying the phrase?

#1 is not true, but I'll answer this anyway. If you call "despite" a preposition (as is very common), then "the fact that I was tired" would be its "object" (or "complement"), and the entire structure ("despite the fact that I was tired") would be a prepositional phrase.

Or is it 1 whole dependent clause because it is subordinate to the main clause? This textbook (image pasted below) seems to say that it is categorized as 1 big dependent (aka subordinate) clause? Is this true?

Most people would not consider the entire structure to be a dependent clause, because a clause generally consists of both a subject and a predicate. As I noted earlier, the last part of it ("that I was tired") would usually be considered a dependent clause. That book seems to take an atypical approach.