As President Kennedy used it (in an imperative), it’s very literary — today we would say “don’t ask”. However, as DavidK points out, it is normal (obligatory) to use “not” instead of “don’t” in the negative infinitive. Additionally, we use that pattern in the subjunctive mood, for example: “It’s important that he not be late” or “it’s important that he not work to hard”. (Note that this refers to a situation that is uncertain and that this conservative use of the subjunctive is more common in American English than in British.)

Use it not in every day conversation. Save it for rhetorical flourish, poetry, etc.

It depends. Typical, (verb + "not") is very formal and not used much in everyday conversations. Ex. "Give not to yourself, but to those around you." "Ask not what it is, but how it is." Those are some random examples, and while yes they make sense, it sounds very formal. BUT! There are exceptions though: "do not..." "have not..." "will not (verb) ..." "should not (verb)..." Those are all veryyyy common in everyday speech.

“Size matters not. Look at me. Judge me by my size, do you? Hmm? Hmm. And well you should not. For my ally is the Force, and a powerful ally it is.” - Yoda (Star Wars) May the force be with you my fellow Jedi Nobu 🥋

The English language has evolved over the years, as is evident with this form of speech, which was once common in earlier periods of time. Nowadays, if one is knowledgable about quotations, they may be used in order to make a point (or a reference) to something by repeating a famous line such as this "Ask not what your country can do for you - ask what you can do for your country" (J.F. Kennedy inagural address).