This is like asking how many words are in the English language. Everyone is different, words are created everyday (and fade away slowly), so naming a specific number is misleading. But for a single idiom, you can compare it’s frequency of usage in the US vs UK using corpora (COA/BNC or Google NGrams)
First a general observation: for questions of these sorts that involve all of a set (however well defined), it will always depend. A particular idiom may be one in the US and not in the UK, and the other way for another. And everything is vague, with degrees of acceptability.
But to answer your explicit questions directly:
- (The main question) 1. How many British and American idioms are either very similar or identical? Is it possible to give a rough estimate or percentage?
Answer: This is a terribly broad question. The concept of a word is already slippery, an idiom even moreso, especially with figurative meanings. How many words are there in the English language? There’s no good quantitative assessment of that. The OED says it has 171,000 words. Are all of those in every ‘English’ speaker’s head. It is said that most adults have vocabulary of 20 to 30 thousand. Not everybody recognizes or uses the same vocabulary. Even though the number of idioms should be much less, what is an idiom in AmE and what is one in BrE needs lots of interviews with individuals to ask them what they recognize and what sounds normal to them, a daunting task. This task has so much variability and vagueness involved that any kind of percentage estimate will be meaningless. Now that’s for the entire (not well defined) set of idioms in both varieties.
Or more mathematically speaking, you are asking: what is the set of American idioms, what is the set of British idioms, what is the intersection of these two sets. And the response is that these sets are too vague to define precisely; membership of any particular idiom in any of these sets is probabilistic (even for an individual speaker they are fuzzy).
So, no, there is no way to give a percentage because defining idiom, defining British vs American, and then counting are all so vague and problematic that math just doesn’t apply.
- Question 2: Is it OK to use an American idiom in a British setting?
Answer: Assuming that you know that an idiom is American and not British (and these labels are pretty slippery), it’ll be acceptable if it is recognizable. That’s sort of empty because what does acceptable mean other than that it is recognizable. I don’t think you’ll be thrown out of English high court for it (but a bunch of old white dudes in wigs might laugh at you or you might never be allowed to see the Queen again).
If no one recognizes the idiom then it’s probably not ‘OK’ or rather it will just be weird since no one will understand you.
As someone else said, American media is more pervasive than British so (in a very vague sense) more idioms with specifically American origins will be understood in the UK than the other way round.
- Question 3: Are there any characteristics that help a learner to recognize the origin of an idiom?
I’ve been giving the impression that there’s no hope here but that’s not really the case. I’m sure there are numerous websites and books about explaining American idioms to the British. But the provenance of an idiom is usually difficult, and each one needs its own separate research, often answered in specialty articles in print or online (ELU is in spirit an attempt to have a place where that sort of thing can be asked about a specific idiom).
But if you’re given an idiom and asked at gunpoint if it’s more likely to be British or American, there might be some heuristics, but I doubt it.
- Question (list): How can I tell which of these idioms are American, English or Australian in origin?
There are countless dictionaries of idioms in English in print and online. Sometimes they tell you that the idiom is primarily AmE, BrE, or AusE, or sometimes not. Or the volume will be about American idioms in particular without conceding itself if it is recognized in the UK. You’ll have to check those books.
In the end, for your first three questions, there is no general comparison, but one can certainly find out something about individual idioms.
About individual idioms there is hope though. There are some things you can do online quickly to tell if an idiom is more common in the US or the UK (sadly I can’t find anything online for other varieties of English so you’ll have to pick up a book on Australian idioms if that’s your thing).
One could guess that the more popular an idiom is in a corpus, the more likely its origin is there. But in some sense that doesn't matter because you probably really just want to know if it will be understood there and the relative frequency is exactly that.
But how to determine relative frequency of an idiom? There are the two websites, COCA (a corpus of US English) and BNC (for UK English). You can search your idioms on both and figure the relative frequency in each (they both give absolute frequencies, so they are incomparable, but if you divide by the number of entries, then you can compare. These corpora though separate are well studied and curated so are high quality. They are only written records, not speech.
Another possibility is Google NGrams. There is a search feature that allows you to track a phrase for a given corpus, and they have an American and a British corpus. An example is
For convenience, I have made the links for your specific requests:
Of course, as with any corpus search, you have to beware of a few things.
false negatives - leaving out things you should have searched for (I modified some of your items to be what I think will catch better) for example, I think ‘belly up’ appears much more likely without the hyphen. leo spelling variants (ax or axe?) can mess things up.
false positives - catching things you realize you don’t really want. Suppose you just looked for ‘belly up’. It might include also the common idiom ‘belly up to the bar’.
OCR - a lot of the text in the Google corpora is scanned and converted to text using optical character recognition. For lots of print (mostly older but still some recent) there are lots of mistakes, so there’s lots of room for capturing or missing unintended things.
how things are said - what you think people write is not necessarily how they write. ‘give someone the ax’ is a good template but no one will ever write that. ‘give him the ax’ (probably not ‘give her’ or ‘give them’ because it feels to me like an older expression and only recently would language show the latter.
There are countless other caveats here (not particular to Google NGrams, but show up there more often because of the source text). The best defense against these systematic errors is to look at the actual hits and correct your search accordingly to include or exclude.