I agree. This is a better translation.

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").

Mindfulness is often translated as Achtsamkeit

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.