Suppose I have the sentence: "This is a website done right."
It sounds wrong, but I cannot find any grammatical rules that confirm my suspicions. Am I just crazy or is that improper grammar?
According to M-W.com, right (adverb) can mean "in a suitable, proper, or desired manner", and gives as an example of that usage "knew he wasn't doing it right." So I'm gonna have to go with the "you're just crazy" option. :-)
According to the Wiktionary entry for "right", there is no meaning of "right" that is an adverb meaning "correctly" (however, there are other adverbial meanings of "right"). So, if you trust it as a source, the (pedantic, technical, grammatical) answer is no. However, "done right" is very commonly used in informal contexts in American English to mean "done correctly". I would not say or write "done right" in any formal context.