I have a question about several related notions in algebraic geometry. I am mainly interested in the question "what is the standard notion?" (if there is such). But I also will be happy to know what is the "natural boundary" of the definition.

1. The canonical divisor: I saw a definition for a normal variety of  the canonical divisor as a Weil  divisor (closure of the canonical divisor on the regular set). Is this notion defined in the non-Normal case? Up to what this notion well defined?
2. Quasi-Gorenstein/$\mathbb Q$-Gorenstein variety: I saw a definition for a normal variety that is based on the above notion of canonical divisor. Do those notions require Normality? This looks weird since it would mean that Gorenstein is not necessarily Quasi-Gorenstein or $\mathbb Q$-Gorenstein (as it is not necessarily normal)
3. (log-)canonical/terminal, Fano, Calabi-Yaw, General type variety: I saw a definition for a normal variety that is based on the above notion of canonical divisor. However it uses the word "ample" which I do not understand when it refers to a general Weil divisor . Does this mean that these notions are defined only for $\mathbb Q$-Gorenstein varieties?
4. Do I understand correctly that though the notions of (log-)canonical/terminal and rational singularities have anlaoges for positive characteristic, non of these analogs are considered standard. Thanks a lot