Learn English – the meaning of “his downfall was a mistletoe”

Yes, it is a familiar, colloquial expression that is used as a substitution for "How are you?" or "How are things?" or "How's it going?" or any of dozens of possible expressions.

Like any such greeting — cf. "How's it hanging?" — it is not meant to be taken as a literal request for specific information. If you are addressed in such a manner it is usually enough simply to respond that you are fine (this can be true even if you're not fine — bland, non-committal greetings such as this should not be viewed as an invitation to go into great detail about your problems).

It would appear that you underestimate the cultural significance of the Rolling Stones!

A quick search for the phrase's history (via Google nGrams) shows that it was not an idiomatic phrase before the song was released in 1966: except for a Southerner opposed to civil rights in the 1920s ("We will paint this State red before we paint it black") and a reference to national colors ("It would be just the same if Ireland began to paint the map green or Montenegro were to paint it black"), (almost) all the pre-1966 instances I find are literal references to actually putting real black paint onto things. It's only after the song became iconic that "paint it black" became an idiomatic expression.

All of the imagery in the Stones song is to death and its accompanying sadness; black is the color of funerals in England, and at least one verse ("I see a line of cars and they're all painted black / With flowers and my love both never to come back") is an explicit reference to a funeral cortege. Using "Paint it black" as the fire-at-will signal is sardonic, and makes excellent television - but is probably not in common usage. (I haven't watched the episode yet, so don't know who was giving that order - if it was, for example, a police/SWAT commander, use of a command like that would likely result in suspension from duty, since the police are supposed to be preventing funerals rather than causing them. A military commander in a war zone might get away with it. It sounds, however, like something the "bad guy" would say.)

It's a pop-culture reference which has become an idiom; the characters in "Elementary" are assumed to know the song and understand its meaning. (Most people in North America and the UK, between the ages of 30 and 65, have heard the song at least a dozen times.) To people from elsewhere, or from a different generation, it has no obvious meaning and should probably be avoided if you wish to avoid misunderstanding.