What would a stationary observer A see when he looks at 'a person B moving at near-light speed holding a mirror in his hand.' I know that B wouldnt be able to tell he's at near light speed and he will see his reflection normally, but what will the observer A see? Would the mirror appear black to A or will the reflection take longer than normal to become visible for him? (Consider the observer A uses an instrument that makes the image visible to him despite Doppler shift). Also, would A observe any difference in the mirror's reflection (except Doppler shift) if B isn't moving at all?
Would the mirror appear black to A or will the reflection appear after a delay?
The reflection would appear after a delay. If the mirror moves at a speed infinitely close to $c$, this delay would approach infinity as seen by $A$. You can say that the mirror appears black only for when the speed of the mirror is exactly equal to light speed, which is impossible.
However, one thing is slightly strange here: The classical reflection law is no longer applicable from the viewpoint of $A$. Indeed, the relativistic reflection law implies that the incident angle is not necessarily equal to the reflected one. Therefore, it is possible that the image becomes blurry, or a flat mirror behaves as a concave or convex one as observed by $A$.