How do you treat an emoticon at the end of a sentence?
He probably caught his cold from the kids at school :(
Should there be a period after the :(, or should I go straight into the next sentence (capitalizing the next word)?
Emoticons are not used in formal writing, so how they are punctuated is really just a matter of style choice. It depends on how you conceive of the function of emoticons.
We can think of emoticons as a parenthetical expression of emotion that helps the reader to understand the text (e.g., "(I feel sad.)"). In this sense, they are rather like stage directions in the script for a play.
Since emoticons are extraneous to the meaning of the sentence and can stand on their own, I would suggest only punctuating the sentence proper.
This would be roughly the equivalent of
If instead you feel that an emoticon instead represents terminal punctuation (e.g., a sad exclamation point), as some commenters have suggested, then you would leave off the period altogether since further punctuation is not necessary.
In any case, I think it would be improper to put a period after the emoticon.
It entirely depends on the writer. Some use emoticons as punctuation, some don't.
However, I have never seen sentence punctuation after a smilie except when parenthesis (or other brackets) are being used, so I can confidently assert that doing so is descriptively unacceptable even if there are not yet any prescriptive rules.
In my experience, these are acceptable:
while these are not:
and this is acceptable, though potentially confusing as the double-parenthesis smilies have additional meanings: