It's used here in Colorado and neighboring states, and probably most if not all of America, but I don't know where in Canada or elsewhere (UK, Australia, NZ, etc.), just that it's used somewhat in the UK and Canada. As others have stated, many people don't even know it's German and very few know what it means. I'd imagine more people in the UK (because of proximity) and New England, especially Pennsylvania (because of many German and Dutch descendants) know what it means.
Contrary to many of the other replies, I believe there is a subset of people who say gesundheit as an attempt to wish the sneezing person well while divorcing themselves from the inherent religiosity of the phrase "god bless you". I have known several people who justify their use of it this way, including myself. That's not to say that what you said isn't true, but just to add another reason why people might use the word.
No, not that I know of a region where it would be used more or less ( native speaker here). To be super grammar nazi: 'Verzeihung' used in the way of 'excuse me' is already a shortened form from 'jemanden um Verzeihung bitten' as in 'Ich bitte Sie um Verzeihung' which means 'I ask you (polite form) for forgiveness'
In the UK at least, Pardon? on its own is most commonly used to imply one didn't hear, and request it be repeated. Pardon!? / Pardon me!? Did you just call me a... can also be exclaimed about something one doesn't like the sound of.
Pardon me as a statement would usually be more appropriate in response to doing something by mistake that might be deemed rude, like interrupting, belching or farting.
Excuse me can be used for either of these too, but would be the most appropriate thing to say if asking to get by people in a busy place. If you said Pardon me, people might think you had just farted or something!
verzeihung mich is wrong. you might say verzeihen sie mir (pardon me) or ich bitte um verzeihung (i beg your pardon). but you should know that verzeihung itself is a rather formal expression, more likely to be used by older people or in written language, and these sentences are even more so.
Thank you - again Leo translates to forgive as jmdm (etw Akk.) vergeben / verzeihen / etw entschuldigen
If I did something really bad, such as forgetting my wife's birthday, which verb would be most suitable? I am guessing vergeben
If you step on my foot, you say: Verzeihung! or Entschuldigung!
"Vergeben" or "Vergebung" says often a Priest.
I can forgive (vergeben) someone who shows no remorse.
I can excuse an act. (verzeihen) A human can I forgive (vergeben).
If I forgot my wife's birthday, then I would ask to "Verzeihung" and I would buy her a beautiful bouquet. ;-)
By the way: correct is: Ich bitte um Entschuldigung oder Verzeihung. But not: Ich entschuldige mich. But then you apologize yourself. (reflexiver Gebrauch - reflexive use).
'I beg your pardon' isn't accepted, although given the translations on Pons, I don't see why it isn't... https://en.pons.com/translate?q=Verzeihung&l=deen&in=ac_de&lf=en
Because that's not what Verzeihung means.
As a noun on its own, it means "forgiveness" or "pardon" (from (jemandem) verzeihen "to forgive (someone)").
As an expression, it's short for ich bitte (Sie) um Verzeihung "I beg your pardon; I ask for your forgiveness" and is thus used similarly to "Sorry" or "Pardon" or "Forgive me" or similar phrases. Perhaps even "Apologies."
But singular "apology" is not a good equivalent -- it's neither a literal translation of the stand-alone noun nor, to my knowledge, a common phrase used when asking for forgiveness.
No, "Verzeihung" is a noun, not a verb, so saying "verzeihung mich" would be like saying, "apology me" or something. It doesn't make any sense. There is a verb, "verzeihen", so you can say, "Verzeih mir!" (informal) or "Verzeihen Sie mir!", but I think that has more of the meaning of "Forgive me!" My understanding is that for a crowd, you would perhaps say, "Entschuldigung!" or "Entschuldigen Sie, bitte!" But I'm not entirely sure, so someone correct me if I'm wrong.