No, darkness doesn't cast a shadow.

At least, it doesn't without a particular DM making it a table rule — which would be a totally reasonable ruling to make. To tackle the question though, we must consider the default baseline from which such rulings would be made, avoiding assuming rulings that give us a result from circular reasoning.

Darkness not casting a shadow is non-intuitive, but magical darkness is inherently non-intuitive and operates by its own idiosyncratic rules.

I've emphasised the relevant parts of its effect description (PBR p. 86; PHB p. 230):

Magical darkness spreads from a point you choose within range to fill a 15-foot-radius sphere for the duration. The darkness spreads around corners. A creature with darkvision can’t see through this darkness, and nonmagical light can’t illuminate it.

The stipulation that it's effect is limited to a radius of 15 feet, and that its effect is to prevent illumination, not penetration of light, means that it does not and cannot alter light levels outside its area of effect.

Contrast this effect description of darkness with that of fog cloud: the latter's description simply says that it creates fog in a given area, and then leaves it as an exercise for the reader to extrapolate from our shared understanding of real-world fog — such as the fact that fog impedes the passage of light — with the help of the game's specific rules for obscured vision. Darkness, by contrast, does not try to leverage our intuitions about "magical darkness" — as none can be expected of us¹ — and instead gives specific effects. These effects are sufficient for most groups, so that groups who don't care (or who prefer a non-scientific fantasy experience) can get on with the game without having to tackle the question "but how does it work?!" with lengthy deliberations first.²

It could be argued that the line "A creature with darkvision can’t see through this darkness" means that light is entirely prevented from passing through. However, such an extrapolation would be a ruling limited to the ruling DM's game, as other DMs will not all also make the same ruling. (Since how darkvision operates is idiosyncratic and non-lawlike in the same way as how magical darkness does, we cannot reason from how it operates to how darkness operates, without a DM adding even more interpretive rulings.) Since such a ruling is an addition and not universal to all possible DMs, and because magical darkness is inherently illogical, that extrapolation can't be considered the default, baseline case.

^{1. Those of us who have a specific metaphysical reality in mind that does create intuitions about how magical darkness operates are precisely those who would want to make rulings about darkness in order to satisfy our intuitions and to make the spell better match our imagined setting. This is exactly what the permission to houserule, that the game gives DMs, is designed to accommodate. For the rest of us, the game doesn't assume anything, and gives us bare effects for things that are not transparent to the usual intuitions of modern humans, like darkvision and the darkness spell.}

^{2. In my own games, I'm likely to rule that it does cast a shadow because I tend toward classicism in my campaign metaphysics — if the issue ever comes up, which I expect it likely won't. Except that, if I'm running a romanticist campaign based on my ontological Chaos premise, I'm likely to say that it doesn't cast a shadow and neither does its contents, for no reason other than "because Chaos." So, there you go: it very much depends on the DM and campaign.}

No, you can't cast and maintain another concentration spell at the same time

The description of the Shadow Magic sorcerer's Eyes of the Dark feature (XGtE, p. 51) is below:

When you reach 3rd level in this class, you learn the darkness spell, which doesn't count against your number of sorcerer spells known. In addition, you can cast it by spending 2 sorcery points or by expending a spell slot. If you cast it with sorcery points, you can see through the darkness created by the spell.

Nothing in that indicates that the normal rule of concentrating on only one spell at a time is suspended when you cast the darkness spell by either method. If you cast another concentration spell, your concentration on the darkness spell will end.