Depends. It can also mean "being aware (of something)": "(sich etw.) bewusst sein". In that sense I guess mindfulness could be translated this way (in case you're wondering, the German term for mindfulness is just "Mindfulness", just like the German term for computer is "Computer").

I agree. This is a better translation.

I'd go a bit more with "being (self-)aware". Being conscious sounds a bit more like a (positive) medical condition, compared to being unconscious, while awareness is more of a mindful thing.