An aside to begin with: these were fully explained in the sources you gave. If they are confusing, please ask about how their presentation of their etymologies/their logic, etc., is confusing to you, rather than just asking about the etymology itself. It saves everyone confusion and repeat questions.
For your first question: Greek τρόπος has a much more general meaning than what you seem to think it does; all dictionaries in Perseus agree that it is something like 'a turn, direction, way' with pretty straightforward expansion into 'a method, manner, style'. The clearly related verb τρέπω 'I turn' also has to do with turning. Textual evidence clearly backs this up as well, with usages for something like the directional 'way' (the LSJ cites Herodotus 2.108 who uses this sense in that passage).
I'm not sure why the Etymonline jumped to the 'musical' meaning. Perhaps a typo? There certainly was a musical meaning (like 'style (of play)'), but it would be much simpler to keep it in the realm of 'writing' and 'argument', and just have it 'figure of speech' all the way down.
Another aside: Greek and Roman scholars, when contemporaries, had excellent command of their own language, of course, but also the other language. This bi-directional language contact situation is a very different than a more one-directional language contact situation.