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comment The free group $F_2$ has index 12 in SL(2,$\mathbb{Z}$)
If someone is still interested in an explicit homomorphism: there is one in Example 2.5 of these notes by Keith Conrads: math.uconn.edu/~kconrad/blurbs/grouptheory/SL(2,Z).pdf
comment Rock-paper-scissors…
I guess "classification" is a soft term. For me, a classification is a one-to-one correspondence to a set of data together with a dictionary that allows you to just read off the answers to many interesting questions about the original objects, or at least answer them easily. In non-trivial cases, a classification will never enable you to answer all questions easily, so one has to specify "interesting questions". In this case "what is the number of instances of a given order?" should be one and if it's the only one, then classification in the sense above reduces to enumeration.
comment Properly Discontinuous Action
Of course, if a torsion-free group acts properly discontinuously, then it acts freely. So in that case the definitions are again equivalent.