You are right concerning 块 (not 快) and 元 being used interchangably for money (although 块 is colloquial). But unless customs changed since I last visited China, people don’t say 九十三分 because the currency is not two-tear like most Western currencies but three-tear 元-角-分 (with each smaller unit being worth 1/10 of the next-bigger one). So it's 九毛三分 (or formally 九角三分).
Kuài is the measure word, it literally means something like a "peice" but is used for units of currency. Yuan is the name of the currency. I'd say 99% of the time people use kuài when speaking. Yuan is only used when written, a bit like a $ sign. There's not really a good analogy for measure words in English, but the closest is something like if you ask for "five bottles" in a conversation you don't always have to say bottles of what because it's implied.