Punctuation, Quotes – How to Punctuate and Continue a Sentence After a Quote

There are several alternatives if you want to retain the quotation marks and question mark, which I list here in shortened form:

1. Before you walk into a room and ask, "Who died?", you should ...
2. Before you walk into a room and ask "Who died?", you should ...
3. Before you walk into a room and ask, "Who died?" you should ...
4. Before you walk into a room and ask "Who died?" you should ...

The basic form is "Before you do X, you should do Y," which means that the comma before you is necessary to separate the clauses and (3) and (4) can be rejected.

Version 1 has a comma before the reported speech; that's standard form. Here, though, it is probably not necessary because the "Who died?" is so closely tied to the first half of the sentence that it doesn't need to be set off from it. There is also the possibility that the first comma could be interpreted as the end of the first clause [whereas it is in fact the second comma], or that the quotation between the two commas is a parenthetical insertion.

I prefer version 2.

If the quotation ended in a full stop and not a different mark, then it would suffice to omit it altogether and simply use the clause-separator:

Before you walk into a room and say "You look sad", you should ...

It is worth considering a further alternative:

Before you walk into a room and ask who died, you should ...

This doesn't use reported direct speech, it reports the activity and describes the situation. It's left to the reader to imagine the scene and the intonation. Do you really need to put that in the writing? Perhaps this is straying into the territory of writing advice.

I agree with Edwin Ashworth that running the quotations as a series of parallel clauses in a single sentence does much more harm than good to the coherence of the final product. Instead of doing that, I recommend getting rid of the "et cetera" at the end of the original sentence and running each quotation as a stand-alone example:

The worst scenario is when your friend refuses to bear any responsibility for the cancellation: “We weren’t really sure we were meeting tonight, were we?” “Good thing I see you all the time anyway.” “I know you understand how demanding my job is.”

If you insist on retaining the original one-sentence approach, I advise punctuating it as follows:

The worst scenario is when your friend refuses to bear any responsibility for the cancellation: “We weren’t really sure we were meeting tonight, were we?”; “Good thing I see you all the time anyway”; “I know you understand how demanding my job is”; et cetera.

The first time you use a semicolon in that series of quotations, you establish the semicolon as the standard mark for binding each of the parallel pieces in that sentence to the others.

Your argument that including a question mark at the end of the first quotation justifies including a period at the end of the second quotation (and a comma at the end of the third one) ignores the special status of the question mark as an unusual end-of-sentence marker, whereas a period is the default (and assumed) end-of-sentence marker. That special status helps explain why a comma can replace a period at the end of certain quotations ("It's too confusing," he said) but can't—or at least usually doesn't—replace a question mark ("Is it too confusing?" he asked).