In the cases you mentioned above, "of" is always used. "That" and "about" are other word that can be used after "reminds me", but "reminds me that/about" has a slightly different meaning to "reminds me of". Examples: "The near-death experience reminded me that life is short." "The near-death experience reminded me about the brevity of life."

Not always, depending on the structure the sentence. Cherry's answer provides some other words that work. Just as in your question about "ravishing", you could write: That reminds me, what did you do about that problem? That reminds me, will you be coming over later?

yep "of" is always added after "reminds me" ^_^