The tooltips are not "contextually accurate"; they don't consider context at all. They are the equivalent of a dictionary, with meanings sometimes wildly inappropriate (e.g. verb meanings when it's noun usage), and not much more. Frankly, that fact is such a source of confusion on Duolingo that I think they either need to get to work on making it contextually aware, or make it more explicitly clear what the hover text is and isn't.
As for how it's possible to know that, common sense rules the day. Where common sense is no help, then trial and error. Sometimes you just have to get it wrong and then engage in the discussion like you are. As for learning the differences on your own, I find Wordreference.com to be excellent at breaking out a word's different contextual uses.
I completely understand about what you're saying about the drop down options being confusing for some, but I hope Duolingo never gets rid of them because I love these drop down menus. When it's teaching me a new meaning for an old word, I can still look and verify the old meaning with the drop down menu or I can learn odd meanings for words I wouldn't otherwise know for years maybe. I hope they don't take this away. I thought it was pretty obvious after a couple weeks of using duolingo that these are just random translations and have nothing to do with context, but maybe they should write that somewhere. I think the drop down translations are one great thing Duolingo has over Rosetta Stone, so I hope they aren't removed.
Doux/douce has slight differences in meaning depending on the noun it modifies. Some things that are doux/douce (soft) could include: cloth, hair, skin, light, music, voice. So you could say "elle a un peau doux" = She has soft skin. But that is her skin, not herself. For people (and animals), doux/douce = gentle/sweet or even in the sense of gentil(le) = nice.