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.
He probably caught his cold from the kids at school. :( I hope he feels better tomorrow.
This would be roughly the equivalent of
He probably caught his cold from the kids at school. (I feel sad.) I hope he feels better tomorrow.
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.
He probably caught his cold from the kids at school :( I hope he feels better tomorrow.
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:
I'm glad. :)
I'm glad :)
That's surprising news! (But I'm glad. :) )
That's surprising news! (But I'm glad :) )
while these are not:
I'm glad :).
That's surprising news! (But I'm glad :).)
and this is acceptable, though potentially confusing as the double-parenthesis smilies have additional meanings:
That's surprising news! (But I'm glad :))
That's surprising news! (And I'm sad :()
I would argue that the biggest reason not to put punctuation after an emoticon, is so you don't inadvertently modify the emoticon itself. Emogis and other animated emoticons may not be affected, but text-based ones easily can be.
An inane example:
"I hope you have a good day. :) " looks totally normal.
"I hope you have a good day :). " is visually confusing. One might ask, "Is Marilyn Monroe smiling at me?"