Well, if you don't mind making a pact with some powerful entity, you could get 2 levels of warlock. This will give you access to a few spells, some of which perhaps being interesting (both stats and flavor-wise) for your character. But most of all, this will give you access to 2 invocations.
One of these invocations should be Devil's Sight, which grants you the ability to see through any form of darkness, magical or otherwise, for up to 120 feet. This is more than enough to cover the 15 feet of the darkness spell. Cast it on your clothes, and bring terror to your enemies.
While this is a good strategy (generally favored by blade pact warlocks), it has some limitations. While under the effect of the darkness spell, your allies cannot see you. As such, any spell requiring line of sight that they would like to cast on you simply cannot work. Healing you might be problematic, as it requires your allies to be able to touch you. They might know that you are smack in the middle of that sphere of pure black darkness but, once they get inside, it's suddenly not so easy to find you anymore.
These limitations also affect your enemies, however. Any attack from the outside of the sphere should have disadvantage as your enemies, even if they can approximate your position, can't exactly see you to aim properly. Enemies inside the darkness should also suffer from disadvantage when attacking you, unless they possess some form of blindsight or a similar ability. When you attack them, you should have advantage on the roll since you are functionally invisible for them.
The biggest cost of this strategy is the fact that it requires multi-classing. While the lvl 19 and 20 abilities for monk would not really be missed (realistically, few campaigns will reach these levels), it will still set you back 2 levels in obtaining you other core monk powers. Note that doing this would get you 2 invocations, however, and that some of them can be quite interesting for a ninja-esque character:
Armor of Shadows (cast mage armor at will, which might be better than
using your wisdom for AC if its under 16),
Eldritch Sight (cast detect magic at will),
- Eyes of the Runekeeper (allows you to read anything, even languages
you don't know... useful for spying!),
- Gaze of Two Minds (share the eyes of a willing target, more spying!),
- Mask of Many Faces (disguise self at will!), etc...
TL;DR
Sacrifice 2 monk levels and multiclass as a warlock. Gain some minor but potentially useful spellcasting, as well as the ability to see through magical darkness and one more ability of your choice!
It depends on where they are standing
The darkness spell is described in such a way as to give the impression that it is something tangible that spills out around corners, but can be blocked by objects. Think of darkness as not unlike a fog cloud spell.
The darkness spreads around corners. [...] Completely covering the source of the darkness with an opaque object, such as a bowl or a helm, blocks the darkness.
Likewise, darkness engulfs non-magical light - or rather, it cannot be illuminated by non-magical light. This means that magical light can illuminate it.
Light in 5e is described as having bounds. Faerie fire indicates that afflicted creatures shed dim light, which only means that area imposes disadvantage on Wisdom (Perception) checks; attacking into dim light behaves as normal.
This is where location comes into play. The bounds of the dim light must penetrate the bounds of the darkness spell in order to be seen. Put another way, there must be no darkness between you and the creature. You need to check the positions and radius of the area of darkness and creatures affected by the faerie fire.
If the radius of the darkness spell's area completely engulfs the radius of the dim light, they cannot be seen by you from any angle. If they are 5 feet or more away from the center of the darkness, they can be seen from the direction the dim light is from the center. This is sort of mathy, but really no more difficult than normal line of sight. Darkness spills out around corners continuously, so just visualize or draw the circles and draw a line to the center of the dim light.
Who do I have advantage against?
Advantage and disadvantage hinge on who can see whom. In general, you have disadvantage on attacks against targets you cannot see, and advantage on attacks against targets who can't see you. However, when neither target can see each other (such as when a target is standing in darkness and you are not), you have neither disadvantage or disadvantage.
From the basic rules on advantage:
If circumstances cause a roll to have both advantage and disadvantage, you are considered to have neither of them, and you roll one d20. This is true even if multiple circumstances impose disadvantage and only one grants advantage or vice versa.
This means that faerie fire's advantage only has an effect when no disadvantage is imposed. It is important to note that, as explained in the question, faerie fire must be cast at a level greater than 2nd, or darkness will dispel it.
You have advantage against:
- Targets you can see that can't see you. It shouldn't matter the
circumstances in which this takes place (Devil's Sight, location,
etc); if you can determine who can see whom, this will suffice for
advantage.
- Targets affected by faerie fire that you can see
- Targets in any situation in which you have at least 1 source of
advantage and no sources of disadvantage
You have disadvantage against:
- Targets not affected by faerie fire that you can't see and that can
see you
- Targets in any situation in which you have at least 1 source of
disadvantage and no sources of advantage
You have neither advantage nor disadvantage against:
- Targets you can't see that also can't see you
- Targets affected by faerie fire that you can't see
- Targets standing in dim light that you have line of sight to (i.e.
you can both see each other)
- Targets in any situation in which you have any number of sources of
advantage and disadvantage (at least one of each)
This answer hinges on the fact that the darkness spell is dark in 3 dimensions, much like a 15-foot radius of fog or smoke. I believe this is true because of how the darkness spills around corners and can be blocked by objects.
If the darkness spell only "darkens" objects and ground in its radius, then it makes sense to imply that the faerie fire penetrates through it, making line of sight meaningless. Is it a floating black orb, or is it a radius of darkened objects? Again, I say the former, but I would personally leave this up to the DM.
Best Answer
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):
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.