I strongly suspect that that is was coined as a literal translation of the Latin expression id est (i.e.) with the same meaning, because the construction makes more sense in Latin.
Originally, that is (to say) was used at the beginning of a sentence, where that referred to the previous sentence, or between the two sentences, just like i.e.:
The first Roman Emperor was Gaius Julius Caesar. That is, he was the first man who ruled the Empire de facto single-handedly, though he was no emperor in name.
In later use, its position has tended to shift around a bit, but it still refers back to the previous sentence or an idea expressed therein.
Your analysis of it is perfectly fine. You could say it is the subject and the that clause an appositional complement to the subject.
This did not happen until 3 am, (namely) the fact that they returned home.
It is of course a fixed idiom, so that the internal structure doesn't matter a great deal any more. Note that it is used in many languages, like French c'etait là que..., Dutch het was daar dat... = "it was there that...".
Sort of. What they mean is something to the effect of "noun+noun compounds are hyphenated when they themselves form part of a "((noun+noun)+noun)" compound. I guess they may extend this to other types of compounds-- you'd need to read the full article to see. (In "credit(-)card debt"-- it doesn't matter whether you hyphenate it or not-- "credit card" is still essentially a noun, but that's not particularly crucial to our point here.)
Now, that all sounds very logical. But there's just one problem: in actual practice, the rule that they suggest isn't actually followed necessarily. Instead, it's very common to simply write "credit card debt".
Best Answer
In the terminology of CGEL, with the aid of a physiotherapist is an adjunct (meaning, one can remove it and the sentence is still grammatical even when all the words in the remainder are taken in the vary same meanings they had before the removal). More precisely, it is an adjunct of means (as in, 'by what means' was something done). Other sources might call it an adverbial of manner. However it is called, it is realized by a preposition phrase (PP). The head of this PP is the preposition with, and the complement of this preposition is the noun phrase (NP) the aid of a physiotherapist. The whole PP is fronted, though it could also be placed at the end, as in Everything turned out fine with the aid of a physiotherapist.
Note that with the aid of is actually not a syntactical constituent of the sentence. The constituent structure is, rather, this: [With [the [aid [of [a physiotherapist] ] ]. As I said above, the whole thing is a PP whose head is with and whose complement is the NP the aid of a physiotherapist. That NP, in turn, is composed of the determiner the and the nominal aid of a physiotherapist. The head of the nominal is the noun aid. The nominal also has a post-head complement, which is the PP of a physiotherapist (it is a complement rather than a modifier because the head noun licences the particular preposition; aid here cannot be followed by just any PP, but only by PPs headed by a very restricted set of prepositions, such as of. For example, on would not work at all, regardless of what followed it). Finally, this final PP has a complement, the NP a physiotherapist, which consisits of a determiner, the indefinite article, and the nominal, which is the noun physiotherapist.