Learn English – As vs That in this sentence

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.

Phrases such as Of course not can certainly stand alone, both in speaking and writing. But they are not fully-formed sentences of the type: simple, compound or complex.

The term for an incomplete sentence is sentence fragment or just fragment. The Cambridge Guide to English Usage (p317) has this extract in the entry on sentence:

Sentence fragments

In ordinary conversation, the sentences uttered may not be fully-formed like those illustrated in section 1 above (i.e., simple, compound or complex). When taking turns to speak, conversationalists often produce sentences whose wording builds partly on the one before, without repeating every element.

...

The use of sentence fragments and elliptical connections between them makes for highly efficient and highly cohesive discourse.

In your example, Of course not is an ellipsis of the fully-formed sentence:

Of course I am not to blame for the increase in deaths.

In answer to the edited question, a stand-alone subordinate clause (dependent clause) is also called a sentence fragment. The Longman Dictionary of Language Teaching and Applied Linguistics (p522) has this entry on sentence fragment:

(in composition) an incomplete sentence which cannot stand on its own.

For example:

Whenever I try to hold a conversation with my parents about my career is a sentence fragment because it is a dependent clause which contains a subordinating conjunction and should therefore be connected to an independent clause.