Has anyone come across a cheat sheet containing basic properties of the most well-known categories (i.e. does it have (co)products, (co)equalizers, (co)limits, etc?)?
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This book is not exactly a cheat sheet, but it's along the lines of what you're asking for.
Examples in Category Theory by Grigore Calugareanu and Ioan Purdea. http://math.ubbcluj.ro/~calu/BOO-0-14.pdf (Book in progress)
There has been sporadic interest at the nLab in creating a database of categories and their properties. Right now such a project is dormant (and I think in principle we'd be interested if someone wants to take this up -- inquiries may be posted to the nForum), but for the moment see what we have here.
There is some accompanying discussion at the nForum, started by someone who wanted to make this a senior undergraduate project, here.