Venga esa mano
 

Hi,

What is it? Context Reverso usually gives a pretty good idea about most everything, but not now. Is it something old fashioned (Give me thy hand)?

In one context: Let's shake on it! (Let's make a deal/finalize the deal!)
In another: Shake! Put it there! (friendly gesture)

This may be considered archaic usage by some. Make sure you extend your hand first, to be understood where the phrase isn't common.

Is it Present Subjunctive?

It's an imperative.

Interesting. Keeping in mind the two hands involved: which one is ordered?

I've never heard this particular phrase used (as noted, it is antiquated), but would suppose it's the other person's hand.

In Central America, «chocar la mano» was commonly used when I was there, while «dar(se) la mano» was the obvious choice in a more formal setting. The right hand was customarily used, unless it happened to be dirty or otherwise indisposed, in which case the right elbow was offered instead.

The dictionaries I consult also give:
estrechar(se) la mano (las manos)
darse un apretón de manos
apretar la mano (las manos)

All of these phrases express a greeting, rather than deal making.
For that, use «estar de acuerdo (en algo/eso)» or «acordar (algo)».

If a dog has been trained to shake hands, the phrase to use is «dar la pata».

So Give me thy hand is an appropriate, if not literal, translation. It's all about his hand.

Thanks!

As Rusty said, if you tell someone "venga esa mano", you're asking the other person to either shake yours or to do a "high five" gesture with you. This is a celebratory expression, it expresses effusive joy (you may agree on something or simply be very glad to meet someone).
You offer your right hand (which is the convention for politeness) and expect the other person to correspond to the gesture.
By the way, we usually don't say "dame tu mano", because we don't expect anyone would offer us a hand that is not theirs, so in this case, "venga" does not involve a third person's hand, only the one of the person you're talking to. ;)
The use of a third person conjugation, in this context, I think, is a way to avoid a "direct" imperative like "dame la mano", which might sound kind of harsh. I guess it's a way to reduce formality too.

You may also find an even more informal version: "vengan esos cinco". "Those five" are the five fingers of the hand we want to shake. ;)

The idea of an actual third person involved never occurred to me. Neither did the idea of my left hand engaging my right one, or vice versa :) . Just wondered, in case of the imperative, what person's hand I'm addressing: my own or the person's I'm greeting. Both are technically third person singular.

¡Vengan esos cinco! can be readily translated into Russian, by the way. Sounds just as informal. Dame los cinco, literally.

Venir is a verb of motion, and that motion is towards the speaker.

Give me five!