You can say verb +ing +off in these type of expressions. To talk impromptu = to speak without preplanning or rehearsal, impromptu is most often used to describe a theatrical performance or the performance of a musical piece, by a band orchestra or musician singer etc. To talk off the cuff is also without planning. However we most commonly use talking of the cuff in daily language. Talking impromptu wold be used to describe someone who had to give a speech without notice for which they were not prepared. For example if they were called onto the stage at any event. You would say "he/they/she gave a speech completely of the cuff" or "an impromptu speech" Talking off the cuff can/could be used to describe a liar or cheat. 1 = correct. 2 = correct it means do do something or speak without thinking about it. 3 = correct. 4 flying off the cuff = to get angry very quickly almost immediately in reaction to anything/something/anybody.

Impromptu - means it was unplanned, you didn't expect to be talking/speaking. Off the cuff - means you did it or said it without giving it much thought. Both expressions don't necessarily mean you are unprepared to speak or do something. For instance, a car mechanic might be able to fix your car immediately, straight off the cuff, with little thought. "Off the cuff" could be a more casual way of saying "impromptu".

I don't believe the expression "to talk impromptu" is a correct use of the Latin term. You can make an impromptu speech, an impromptu remark, do an impromptu performance. The word impromptu functions more like an adjective. The expression "off the cuff" is a phrase that stands on its own grammatically. It comes from the difference of reading "off the paper" of the prepared speech or "reading off the cuff" without a prepared speech. The cuff is the end of each arm of your shirt. Literally, you extend your arms out to the audience in a body language sense, and talk off the cuff ends of your shirt, instead of holding a piece of paper in your hand. The location word "off" means "from". "He fired off the gun", is another example. So yes, "off the cuff" means literally without preparation. It can also mean without thinking. You cannot really "lie off the cuff" because you are thinking that what you are saying is a lie. If your remark happens to be "off the cuff", you can later find out that it is wrong. That does not mean you lied off the cuff though. You instead said something without thinking if it is correct. If you are good at lying without preparation, then you are fluent in lying. You need a good memory though to remember what lies you have told, and so next time you must remember what you said to be consistent, otherwise you could contradict what you said last time. This needs preparation before speaking, even if you are very quick at preparing.