Usage of /i:/ ("EE-thur") and /ai/ ("EYE-thur") in Great Britain and in Canada seems to be mixed. In the United States, the predominant usage has always been /i:/. However, there's also a long history of /ai/ occurring among a few Americans, including Benjamin Franklin and James Fenimore Cooper in earlier times, and Franklin Roosevelt, Ronald Reagan and Barrack Obama more recently. For many years, /ai/ was associated in the United States with British usage and, by extension, with certain elites who tried to imitate British usage. It seems to have become more widespread in recent decades, however. While some Americans have adopted /ai/, perhaps because they feel that it sounds sophisticated, other Americans regard it as pretentious.
Spelling, by the way, has nothing to do with the difference. In English, the spelling ei usually represents the "long a" pronunciation (IPA /ei/), as in eight, feign, or rein. In such words, it is derived from the Middle English /ai/ diphthong, which normally developed into the "long a" sound. In a smaller set of words, such as receive, ei represents the "long e" sound /i:/. It's rare for ei to represent the "long i" sound /ai/ in words that have been in English more than two or three hundred years; most words spelled with ei and pronounced with "long i" are recent borrowings, such as Poltergeist (from German), or other words that only recently developed a standard spelling, such as heist (originally a variant of hoist).
Contractions generally sound a little more informal than their non-contracted equivalents. However, they also sound more natural, as non-contracted forms are practically never used in speech (except e.g. for emphasis or in cases where contractions are not grammatical).
Whether a particular piece of writing is "formal" enough to warrant avoiding contractions is really quite subjective. If you look at many scholarly books and even journal articles, you will find that many (native speaking) authors actually do use contractions and their respective editors have decided that they're happy with them. I would argue that contractions are almost always possible in e-mails: if the context was that formal, you probably wouldn't be communicating by e-mail in the first place. But as I say, it is a subjective decision.
On the other hand, if you are writing in a formal context such as a journal article or a formal letter to a company and can't decide whether or not to use contractions, then I think that avoiding contractions will always be a "safe" decision in such formal contexts.
Best Answer
I did some searches in the Corpus of Contemporary American English and compared the results to similar searches in the British National corpus.
What I found was that overall, in American English there was a 7.9-to-1 ratio of don’t to do not. With breakdowns by type:
In British English overall, the ratio was 4.4-to-1 in favor of don’t, with breakdowns by type:
So, if it is reasonable to conclude anything from this data, it is that Americans overall use don’t about twice as frequently as the British, but the British use don’t in speech about 2.9 times more frequently than Americans. In any case, these are not big enough ratios to be noticeable by anyone not counting every incidence.