Three questions:
"Appropriate" depends on context and audience, but I know what you mean. To answer your question, there are no uses of shit that would not be considered profanity, and it should not be used in "polite company" in any context.
The most common non-profane substitute for "shit happens" is "stuff happens." It's suitable for general audiences, but everyone will know what you really mean anyway.
I'm one of those crazy people who don't curse at all, unless quoting someone else, and then reluctantly. I generally use "C'est la vie" (Actually French for "That's life.") or "Que sera sera" (Spanish for "What will be, will be," just like in that song...)...With some people I'll use "wyrd bið ful aræd," but I generally translate it to "Fate is inexorable." The last one, especially, might come off as a bit pretentious, but "C'est la vie" and "Que sera sera" are well-known enough, at least in the US, that you shouldn't come off as over-educated or anything.
None of these are as flippant as "Shit happens." If you want to reduce the vulgarity, but keep the same level of flippancy, you could probably use "Crap happens." Most people do use "Stuff happens," though.
It's not "obscene" -- obscenity relates to sexuality, so "fuck" is obscene; "shit" is scatological.
Using a word like "shit" around children will make you unpopular with their parents (though probably popular with the kids themselves). It's accepted in private conversation with most adults.
The phrase "shit happens" is very dismissive of misfortunes, so I would advise you not to use it in reference to other people's troubles, unless you know them very, very well and they can be sure of your actual sympathy. Shrugging off your own difficulties with those words, while vulgar, shows a fine stoicism.
Recently here in Australia the Leader of the opposition got in political trouble because, in response to an Australian soldier dieing in Afghanistan he said (on television) "sometimes shit happens". He wasn't criticised for saying "shit", he was criticised for being flippant about a soldiers death.
I cant speak for other cultures, but here the dismissive nature of the comment is potentially more rude than "shit", a word which is regularly heard from journalists, politicians, academics, etc.
If you said "crap happens" or "stuff happens" people would think you were being pretentious.
In general though the best advise with swearing in a language / region other than your own is don't :)
As I've taught my daughter, if she's not sure if something's appropriate and she has to decide on her own, just think of this question: "Would The Queen approve?"
