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.
I think there are several interpretations, with no definitive rules support one way or another, so it's dm judgement call.
No, flaming sphere stays lit, and pyrotechnics causes the fireworks or smoke
The flaming sphere stays lit and you get the fireworks/smoke.
The reason the sphere stays lit is that it is fueled by magic, not mundane fuel.
While there isn't explicit rules support for this interpretation, it isn't unbalanced because it isn't more powerful than the spells separately, and deciding in favor of the PCs and the rule of cool argue in favor of it.
Perhaps for fluff it goes out during the fireworks/smoke, then relights.
A Contrary Argument
A contrary argument would be that you get either spell, but not both. Basically either one spell or the other wins.
If pyrotechnics wins, then the flaming sphere is extinguished and you get your choice of the fireworks or smoke.
If flaming sphere wins, then pyrotechnics fails to extinguish the flame.
This is making the assumption that the fire in the flaming sphere is magically fueled for the duration, and it seems reasonable to assume that it can't be snuffed the same way as a regular fire, because magic.
So how to decide which spell wins? Assuming they're both cast by the same caster, it seems reasonable to let the caster choose. If cast by different casters in an antagonistic situation, if the spells have been cast at different levels, higher level wins; otherwise, its a contest between spellcasting abilities.
No matter what, the fires created by the flaming sphere are fair game
You can use the flaming sphere to ignite something, and then use pyrotechnics to blow that stuff up like a boss, or, create smoke, of course.
Again, there's not much in the way of rules support, but if you think allowing the sphere to stay lit is overpowered, then perhaps this is the way to go.
Note:
My original answer was significantly different from this but editing it made me re-think it.
Best Answer
Control Flames will work on normal fires started by magic.
None of these spells produces a magical fire. They use magic to start a fire.
You'll notice that all of the things that might be lit by these cantrips have normal fuel (oil, wood, etc.). They are not fueled by magic (unlike Create Bonfire). Magic was only used to create the initial spark. The cantrips above take place instantaneously and are not sustained by the magic or caster.
There are also other fire spells which do not start fires, such as Chromatic Orb (fire) and Flame Strike. (All of these spells are in the PHB except Control Flames and Create Bonfire from XGtE.)
You can expand the fire into an adjacent 5' cube (containing fuel) each time you cast this cantrip.
If you spend 1 action to expand the fire into each ground level cube of the barn, the fire will do the rest. This might even be overkill, but you might as well keep expanding it since you're already standing there watching it (from within 60').
(Control Flames lets you sustain 3 non-instantaneous effects on the fire at once. You don't need any of those.)
Also note that you could just use the Firebolt cantrip (120') itself to start the barn on fire without getting so close.
Control Flames (60') could also be used, but would not be necessary. This would be safer than having to walk up to the barn with a torch in order to cause it to spread.