I hear "come again" is an inappropriate alternative to pardon me or what did you say? Is that true only for certain native English regions or generally not the right way to ask?
I don't know about 'inappropriate', but it is certainly less formal than other ways of asking for somebody to repeat themselves. But I would also consider 'what did you say' as having the potential to cause offence, depending on context. Certainly many older English speakers will have grown up being frequently admonished by our parents: "don't say 'what', say 'pardon'".
In order from most 'polite' to most 'rude', I'd say these are the various ways of asking someone to repeat themselves.
I beg your pardon?
Would you repeat that?
What did you say?
Regional preferences and changes in popular idioms over time mean that there's no universal phrase that is guaranteed to work always and everywhere.
However, you can avoid these kinds of problems by not using idioms. If you're unsure of the appropriate idiom, you can express yourself in an unabbreviated way: "I'm sorry, I didn't understand that/I couldn't hear you. Would you mind repeating that?"
Having just done a quick Google search, I get the impression nobody really knows exactly how or when this usage first arose...
Come again? (idiomatic, informal)
Could you repeat that? Repeat that please.
- a polite formula used when one has not heard or understood what has been said
Note that the usage is simultaneously informal AND polite. But it's very much a declining usage - as implied by the ELU question About the usage of term 'come again', where apparently because some people (particularly, younger Americans, I think) aren't actually familiar with it as a "natural" usage, they see potential sexual overtones (through the "coarse slang" usage to come = to reach sexual climax).
Personally, I still use the expression myself from time to time. But given the circumstances, I think I'd advise learners just to recognize that it still has some currency, but not to include it in their own "productive vocabulary".
It's difficult enough for learners to appreciate the finer points of "informality" and "politeness" at the best of times, and usually any given usage only falls into at most one of those categories. Given this one falls into both, it's simply not worth trying to figure out when and why you might choose to use it, rather than going with either "obviously informal" Eh? What? or "obviously formal" I'm sorry, I didn't catch that. Could you repeat it, please?
The Oxford Dictionary simply says
Used to ask someone to repeat or explain something they have said.
In BrE it is used in colloquial speech, but it can be said in a way to imply disbelief. For example
I just gave away our dog.
I would not say "come again" at a business meeting.