I'll preface this by noting that I don't disagree with anything you've said, but I do have some comments:
> Similarly, “j’ai un chat dans la gorge” probably translates best as “I’ve got a frog in my throat”, even though it’s a completely different animal, it’s an obvious mapping.
Those obvious mappings can sometimes be too seductive for the translator's good. One example is that people translating English-loanwords-in-a-foreign-language into English usually can't help but translate them as the original English word.
Another example is that, in China, there is a cultural concept of a 狐狸精, which you might translate as "fox spirit". (The "fox" part of the translation is straightforward, but 精 is a term for a supernatural phenomenon, and those are difficult to translate.) They can do all kinds of things, but one especially well-known behavior is that they may take the form of human women and seduce (actual) human men. This may or may not be harmful to the man.
Because of this concept, the word also has a sense in which it may be used to insult a (normal) woman, accusing her of using her sex appeal toward harmful ends.
Chinese people translating this into English almost always use the word "vixen", which is, to be fair, a word that may refer to a sexy human woman or to a female fox. But I really don't feel that they're equivalent, or even that they have much overlap. (Unlike the situation with English loanwords, I think native speakers of Chinese are much more likely to choose this translation than native speakers of English are.)
> what about for example “C’est pas tes oignons”, which literally means “these aren’t your onions”
The form closest in structure to that would probably be "none of your beeswax", which is just a minorly altered version of "none of your business". I assume the substitution of "beeswax" is humorous and based on phonetic similarity.
As you note, there are multiple dimensions relevant to translating this and several positions you could take along each. For this particular idea, I would say the two most important dimensions are playfulness and rudeness; it's a very common idea and the language is rich in options for both.
> translations often vary in terms of how faithful they are to the source vs how idiomatic they are in the target language. Take for example the French phrase “j’ai fait une nuit blanche”, which literally means “I did a white night”. Clearly that’s a bad translation. A more natural translation might be “I pulled an all-nighter”.
This isn't what I had in mind. Here are some idiomatic translations:
I pulled an all-nighter.
I was up all night.
I didn't get any sleep.
I never got to bed.
I've been up since [something appropriate to the context].
[Something appropriate to the context] kept me up all night.
I wouldn't call any of the first four "more paraphrased" than the others. (The last two might be, if they included extra information.) If these were reports of the English speech of some other person, one of them (or less) would be a quote, and the others would be paraphrases. But as a report of French speech, they're all paraphrases. The first shares a little more grammatical structure with the French, which doesn't really mean much.
For a fairly similar example from my personal life, someone said to me 这是我第一次听说, and my spontaneous translation of it was "I've never heard that before", despite the fact that there is technically a perfectly valid English expression "this is the first I've heard of that".
What's closer to the grammatical structure of the Chinese? That's hard for me to say. You could analyze 我 as the subject of 听说, and I lean toward that analysis, but my instincts for Mandarin are weak. You might see 我 as being more strongly attached to 第一次, meaning something more like "my first time (to hear ...)" than "I hear (for the first time) ...".
But for whatever it's worth, a word by word literal gloss would be "this is me first time hear".
Between languages with less historical interaction than English and French, it's quite possible that a syntax-preserving translation of some sentence just doesn't exist.