1. Yes, you can use it without inversion, but 99% of the time people will put “Under no circumstances” at the beginning of the sentence (both in written and oral English). So, it sounds unusual to put “under no circumstances” at the end. That said, it sounds perfectly natural to put “under some/certain circumstances” at the end. So, the oddness of not inverting the sentence seems to be limited to “under no circumstances.” 2. Yes, inversion of this phrase is commonly used in spoken English.

Ofcourse you can omit the inversion but that can change a main point in meaning. For example " I cannot come until Friday" sounds neutral negation but "Not until Friday I can come" a speaker emphasised that the Friday is crucial in the sentence. ☺️

1. Strictly speaking, no. But you can still express the same idea without inversion. In some cases, you can move the negation around from one clause to the other, and then switch the clauses themselves around. Except maybe in "no sooner... than" , and perhaps in some other cases it could change the meaning, but I really don't think so off-hand. The following examples show correct (in my opinion) examples of inverted and non-inverted sentences. Inverted: "Under no circumstances can you use your phone here." Non-inverted: "You can't use your phone here under any circumstances." Inverted: "Not until you've eaten supper, will you have any dessert." Non-Inverted: "You will not have any dessert until you've eaten supper." This last one is a little tricky, since I think you need to replace "sooner", and I don't think "later" would fit. Inverted: "No sooner had I finished than the alarm rang" Non-inverted: "The alarm rang just as I had finished". (Or close enough in my opinion.) 2. There's no reason you can't, but it can sound rather official, literary or perhaps even a little old-fashioned. But you could always use it sparingly for effect, especially if you're telling a story. That's just my own personal opinion anyway.