As a person from the US, the "correct" spelling of neighbor doesn't have a u in it. Period. :p

In a somewhat related way, I can usually tell within a few sentences if a person is a non-native to the US speaker.

Well, from what I gather, majority of EU learns British English. That's also the one I personally prefer, despite getting frustrated about all of the additional u's. ;) I think that I had once a discussion on Hubski IRC that went something along the lines of:

  Me: yeah, sorry. Had to look up how to write it.
  A: What? Salute?
  Me: That's not my fault! You guys put all the unnecessary vowels into words. For what I know it could have been 'saloute'.
  A: Which ones you think are unnescessary?
  Me: It's just "salut" in Polish

But yes, I can agree. There are even some fairly sophisticated methods of detection country of origin of a non-native language speaker. Here is a full diploma project of a Cambridge CS student titled Identification of a Writer’s Native Language by Error Analysis where the author took lots of written samples and analysed the text with respect to most common spelling errors, missing punctuation, preposition use and tenses of verbs among many other factors. And it turns out to be quite reliable.

While I agree that poor grammar can be a problem that annoys me, I also try to realize that people on the internet are writing from all over the world.

And that's damn great of you, really.

For what it counts I'm trying to write to the best of my ability, as nearly all non-natives I know, but it is oftentimes hard in one way or another. Sometimes you simply have to say 'screw this' and approximate grammar or punctuation, because you realise that the thing you have been writing for past twenty minutes can be done in your language in less than three. One of the things that are the most problematic for me are some of the tenses. See, in Polish it's perfectly valid to say Czytam od rana do wieczora, co robię teraz. that literally means I read from morning until evening, I am doing it right now instead of something like I will have been reading until evening since morning or whatever it's supposed to be. ;) Nonetheless, something that I don't even need to think about in my language turns to be this "what was the tense that describes something that began already, will be happening and you are in the present and relate of a thing that happens? Future perfect continuous?"

Should that affect the local rules for grammatical correctness in formal writing? I'm less sure of the answer to this as time goes along.

No idea. I don't feel like an authority even on things that I actually study, let alone linguistics.

Actually: ThatFanficGuy - would you mind pitching in to this tread? If there is anyone here who happens to have both outsider's perspective and solid level of expertise, it would be you.