Okay, if we're taking them one by one ...
I would understand (but not use) 'Pardon' as a short form of 'I beg your pardon?', which I use with people I don't know well or to be as polite as possible.
I would understand (and use) either 'Excuse me, ...' or 'Pardon me, ...' if another sentence followed, like 'I didn't hear what you said.'
But I wouldn't use either one as a freestanding question, because their default meaning is to get someone's attention politely, so that the other person might think you were about to say something else, not that you needed to hear something again. Also, depending on the tone, alone they can sound a little peremptory and disapproving, implying raised eyebrows, similar to 'I beg your pardon?'
I agree that 'Sorry?' is more common in BE, but I use it myself sometimes because it's shorter than 'What?' but not as abrupt-sounding.
I agree that 'What?' isn't exactly rude, but I also wouldn't recommend it as a default. To me there needs to be a reason for being so short. The mower example is a good one; 'What?' is definitely the most practical reply when neither person can hear well because of distance or loud background noise, so that only short, clear words are audible.
In other circumstances, though, 'What?' (or, definitely less polite, 'Huh?') can imply that there's no need to be polite, that you're deliberately dispensing with polite forms. Among the people you see every day, and with whom you talk very casually, that may in fact be perfectly reasonable a lot of the time, like when you're doing something quick or talking about something else, and no one really cares.
But if you always say only 'What?', even family, friends, or co-workers could get the impression that you're taking them for granted or can't be bothered to use ordinary courtesy words, like falling out of the habit of saying 'Please' and 'Thank you.' True, different families and groups have different conversation styles, so that might not necessarily be a big deal, but it's something to think about.
'What?' also often seems less polite and more impatient to me if it comes across as implying that the other person should have spoken up more loudly and clearly. That's okay if the speaker really was mumbling, or should have realized that no one can hear over a lawnmower. But in practice, the hearer may not have been really listening or paying attention, or may be deaf and choose not to use a hearing aid, in which case saying 'What?' or 'Huh?' again and again can get tiresome.
'Come again' and 'say again' both sound like they could be regional or class-related. To me, neither one sounds very nice -- again, maybe not exactly impolite, but also not particuarly polite, and not something I would recommend to a learner of English.