Are there any atmospheric condition under which a visible raindrop can fail to hit the ground by evaporating first? I imagine this would require a large vertical temperature difference, and possibly the rain forming very high up. Has anything like this been observed experimentally, or if not - is it possible to perform a calculation to show whether this is plausible?
Yes, most certainly, and meteorologists call this kind of rain Virga (see Wikipedia page of the same name).
These are the salient and more interesting points of the Wiki article: