Learn English – When quoting a quotation, how do you handle the double quotes

In your first query, the question mark clearly does not belong to the euphemism, so it should not appear inside its quotes:

“Why should I ever hear about this... ‘agency’?”

In your second query, you can dispense with the exclamation mark. While that does go with the word Stop, it isn’t heard and doesn’t need to be quoted. And the final full stop isn’t needed either — you’ve already conceded that right at the start.

He screwed his eyebrows, and asked, “Are you sure you heard ‘Stop’?”

Punctuation is only there to assist understanding. Too much clouds the meaning because although it might be strictly correct, it requires working out! Use enough to make the meaning clear without undue effort sorting through it all.

I don't think I can do better than to quote the Wikipedia page, "Quotation marks in English", which says:

Single or double quotation marks denote either speech or a quotation.

A quotation needs a source. Speech, though, just needs a defined speaker.

There IS an exception for paraphrases described:

Quotation marks are not used for paraphrased speech. This is because a paraphrase is not a direct quote, and in the course of any composition, it is important to document when one is using a quotation versus when one is using a paraphrased idea, which could be open to interpretation.

If Hal says: "All systems are functional", then, in paraphrased speech:

Incorrect: Hal said "everything was going extremely well".

Correct: Hal said that everything was going extremely well.

Seems to me that the rationale given there is incorrect, however. This is nothing to do with accuracy of attribution, and all about how a paraphrase is in a different voice; the pronouns mean different things inside the quotes as out.

So in your example:

Bob: I think I heard him say, "I like you".

...the quotes are appropriate even if it's a misquote, because it's still wrapping speech and assigning the quoted pronoun "I" to the speaker, "him", and not to Bob. If Bob heard right, 'he' likes 'you'. Whereas,

Bob: I think I heard him say, I like you.

... has three unquoted "I"s, all meaning Bob. If Bob heard right, 'he' thinks Bob likes 'you'.

To be a paraphrase in the sense of that exclusion, Bob would need to fiddle the pronouns:

Bob: I think I heard him say, he likes you.

So a more general rule might be: quote marks are required for any phrase where "I" would mean something different to the surrounding text. In which case, both of these are OK:

Bob: I said, "I like you".

Bob: I said, I like you.