How to capitalize trope names outside the wiki?

For convenience, we tend to default to all-first-letters-capitalized, but there's no real need outside this wiki. For example, I prefer to put "big bad" in lowercase, because all pre-TV Tropes uses I could find on Buffy newsgroups and such were written like that. Likewise, there's pretty much no reason to ever treat "deconstruction" as a proper noun. On the other hand, judging from Wikipedia's list of chess gambits, "Xanatos Gambit" should probably be capitalized.

Any other examples of stuff getting capitalized that really shouldn't?

It's a title. If you're referring to it specifically by title, with link to the site, then you may wish to use the capitalization we use. Also if it's a proper name like Xanatos Gambit. However, it's up to you. There's no established rule.