In your example sentence, everything before the comma is a dependent and introductory clause. You could rephrase the sentence by dropping it completely and the sentence's essential meaning would still be retained:
(A popular answer to the common introductory question "What is your favorite color?"), red is a color that is beloved by many.
Red is a color that is beloved by many.
It takes the same type of construction as any other introductory clause:
(Being quite popular), red is a color that is beloved by many.
Both US and UK styling would say that question marks come before the second quotation mark.
US styling would say that commas come before the second quotation mark; UK styling would also say that they come before the second quotation mark only if the comma is part of the syntax of what's being quoted—otherwise they come after.
- US: "There's nothing in the study," he said.
- UK: "There's nothing in the study", he said.
However: "There's nothing in the study," he said, "the living room or the kitchen." Here, the comma comes before the second quotation mark because, without the interjected narration, the sentence would be written as "There's nothing in the study, the living room or the kitchen." In other words, the comma would be part of the quotation regardless. (The serial comma is not typically used in the UK, so I didn't use it here.)
Question marks (and exclamation points) always come before the second quotation mark—even if they don't end the sentence itself.
So, that covers normal situations. Your example, however, is not quite normal because it actually involves the use of both a question mark (inside what is actually being quoted) and a comma (which punctuates the surrounding text).
While the answer provided by Shoe does indeed quote The Chicago Manual of Style (the 17th edition says the same thing), it fails to put it into a specific context.
In fact, Chicago specifically says to omit the comma inside quotation makes in the case of dialogue. It's only in the case of titles of work that this doesn't happen.
Here is Chicago 6.125:
When a question mark or exclamation point appears at the end of a quotation where a comma would normally appear, the comma is omitted (as in the first example below . . .). When, however, the title of a work ends in a question mark or exclamation point, a comma should also appear if the grammar of the sentence would normally call for one. This usage recognizes not only the syntactic independence of titles but also the potential for clearer sentence structure—especially apparent in the final example, where the comma after Help! separates it from the following title. (The occasional awkward result may require rewording.)
“Are you a doctor?” asked Mahmoud.
but
“Are You a Doctor?,” the fifth story in Will You Please Be Quiet, Please?, treats modern love.
All the band’s soundtrack albums—A Hard Day’s Night, Help!, Yellow Submarine, and Magical Mystery Tour—were popular.
In the example given by Chicago, "Are You a Doctor" is in quotation marks not because it's spoken dialogue but because it's the title of a short story. (Book titles are italicized, but articles and short stories are given within quotation marks.) You can tell it's not a quotation because of the capitalization.
Your example sentence is quoted dialogue, and so it follows the first example given by Chicago where the comma would be omitted.
However, in your specific case, the comma also serves an additional syntactical function (that of following an introductory clause) where omitting it is perhaps as awkward as not omitting it.
Personally, I would keep the comma but put it after the second quotation mark—assuming I kept it at all. If I didn't keep it, I would rephrase the sentence so that there was no awkwardness:
Red—a popular answer to the common introductory question "What is your favorite color?"—is a color that is beloved by many.
