The seven coordinating conjunctions (that is to say words that can be used to connect two independent clauses without one becoming subordinate to the other) are for, and, nor, but, or, yet, and so (acronym FANBOYS). Conjunctions like although and because are subordinating conjunctions, in that they make the clause that follows them depend on the rest of the sentence for their meaning.
Conjunctions, whether coordinating or subordinating, can happily be inserted between two clauses with no more punctuation than a comma. Sticklers for punctuation would tell you that this is not true of therefore, which is not a connector between two clauses and which should be preceded by a semicolon or a full stop when it begins a new clause. The word therefore is a conjunctive adverb. You can see that it behaves like an adverb when it occurs in the middle of a clause: "It was cold and I therefore stayed indoors". Here it answers the question why in relation to the verb stayed. You might choose to place therefore before the subject I and it would continue to play the same role, but it cannot replace the conjunction and. You need at least a semicolon: "It was cold; therefore I stayed indoors".
Any expectation of a comma in the examples of the OP has very little to do with the subordinate clauses' restrictiveness, but rather, as the OP suggested, with an interruption of their natural flow. When leading a sentence with a subordinate clause, the comma does not force a "parenthetical / non-restrictive" interpretation. Simply, compare the meaning of two sentences:
- If you work hard, you get rewarded.
- You get rewarded if you work hard.
None of the embedded phrases in the examples were relative clauses, so the concern of imposing a non-restrictive interpretation is irrelevant. In every case, the embedding did put the interrupting phrases in a parenthetical position--even if they are considered "essential" to the meaning of the sentences.
The reference to section 6.32 of The Chicago Manual of Style 16th edition established a legitimate exception to a general rule of commas. If omitting an appropriate comma creates no ambiguity, omitting it becomes a matter of style opinion rather than grammar. Moreover, if we believe an appropriate comma introduces ambiguity, our best solution is to recast the sentence to remove ambiguity.
Considering the options for each example:
1) This is the country where[,] if you work hard, you get rewarded.
(relative clause)
The relative clause is where you get rewarded, and the comma is appropriate, because the embedded conditional phrase, if you work hard, interrupts the natural flow of the clause. The conflict between the locative where and the conditional if might be manageable enough, but many would be more comfortable with the extra comma. If the sentence had been written: This is the country where you get rewarded if you work hard, certainly no commas would be needed.
2) We need to talk because[,] if we don't, we will be in trouble.
(adverbial clause)
The adverbial clause is because we will be in trouble, and the comma is appropriate, because the embedded conditional phrase, if we don't, interrupts the natural flow of the clause. With such minuscule conflict between because and if, there is very little risk of confusion in omitting the comma. If the sentence had been written: We need to talk because we will be in trouble if we don't, certainly no commas would be needed.
3) London is where[,] when I was young, I used to live. (noun clause)
The predicative is where I used to live, and the comma would be appropriate, because the embedded adverbial phrase, when I was young, interrupts the natural flow of the clause. The locative where and the temporal when are nearly irreconcilable and should probably be separated by a comma. If the sentence had been written: London is where I used to live when I was young, certainly no commas would be needed.
4) Give me a call if[,] when you are at the station, it rains.
(adverbial clause)
The conditional clause is if it rains, and the comma is appropriate, because the embedded adverbial phrase, when you are at the station, interrupts the natural flow of the clause. The conditional if and the temporal when seem to be in deep conflict and would work better with a comma between them. If the sentence had been written: Give me a call if it rains when you are at the station, certainly no commas would be needed. The slight ambiguity could easily be eliminated by recasting the sentence to communicate the true intentions of the imperative.
5) It is useful when[,] if it rains, you have an umbrella. (awkward
adverbial clause)
The awkward adverbial clause is when you have an umbrella, and the comma is appropriate, because the embedded conditional phrase, if it rains, interrupts the natural flow of the clause. This construction is awkward with or without the comma, but would probably be less confusing with the extra comma. If the sentence had been written: It is useful when you have an umbrella if it rains, certainly no commas would be needed. The overall awkwardness still suggest a need to recast the sentence.
6) She is the person who[,] if she is faced with difficulties, can
handle them very well. (relative clause)
The relative clause is who can handle them very well, and the comma is appropriate, because the embedded conditional phrase, if she is faced with difficulties, interrupts the natural flow of the clause. The conflict between the relative who and the conditional if might be manageable without a comma, but many would find it less confusing to see the comma. If the sentence had been written: She is the person who can handle difficulties very well if she is faced with them, certainly no commas would be needed.
7) He said that[,] if all goes well, he will call. (noun clause)
The noun clause is that he will call, and the comma is appropriate, because the embedded conditional phrase, if all goes well, interrupts the natural flow of the clause. The conditional in the context of reported speech is the least awkward of the seven examples and fits the exception of The Chicago Manual of Style perfectly. If the sentence had been written: He said that he will call if all goes well, certainly no commas would be needed.
Conclusion:
The ultimate purpose of commas is clarity. Use one if it makes things more clear. Leave it out if it makes things less clear, and in my humble opinion: when in doubt, leave it out. Most importantly, recasting the way we put phrases together can eliminate most of our comma confusion.
Best Answer
श्री गणेशाय नमः
The problem here is that parts of speech (adverb, conjunction, pronoun, etc.) are being used as labels to classify words as "being" one and only one part of speech, and that this categorization seems to be directed by faulty definitions.
Parts of speech are uses of words, and in English almost any word can be used in a number of such ways. Definitions of such use categories are just summaries, and not directions to be followed.
In this case, the wh-words that have adverbial meanings
(temporal: when; locative: where; purpose/cause: why; manner/means: how)
are indeed used to introduce clauses, like many adverbs; but this doesn't make them conjunctions.
Conjunctions are one special class of words with special grammar.
When, where, why, and how belong, however, to a different special class of words, with equally special grammar. In fact, they constitute one subclass of this special class.
Most of the other wh-words (what, which, who) refer to nouns, so they're called pronouns --
either interrogative pronouns (because they're used to introduce question clauses)
or relative pronouns (because they're also used to introduce relative clauses).
These are also the proper terms for the wh-words with adverbial senses. They came originally from Indo-European oblique noun cases like Locative and Instrumental, and they were used in much the same way as any other interrogative or relative pronoun, to introduce clauses.
Since they don't refer to nouns, however, clauses that they introduce have adverbial restrictions.
Relative clauses, for instance, must modify a noun, but if the relative pronoun is where, the noun it modifies has to be place, or a synonym, or some other word that refers to a place. Similarly, relatives headed by when must modify nouns referring to a point or length of time; the time when itself is most common.
Why and how are even more restricted.
How itself cannot be used at all as a relative pronoun, even if it's modifying the right noun:
but not
Why relative clauses must modify the noun reason. Synonyms don't work.
Essentially, the phrase the reason why has become frozen.
There's a lot more, but I forbear.