1) What might they be manipulating/controlling/creating/etc from the perspective of modern science?
The three most important questions I have would be...
- Where are they getting the energy?
- How are they manipulating things at a distance?
- How does this work with human biology?
The Energy
Cold
Cold is reducing the temperature of matter. This is just transferring energy, but transferring it where? And how are they affecting something at a distance?
Energy can neither be created nor destroyed (unless you expand spacetime itself which is what Dark Energy is thought to represent, but let's not go there), so the energy you're removing to make a thing cold has to go somewhere. For small objects this could be dissipated into the surrounding air and ground providing a limit on how much they can make how cold.
Alternatively it could be accompanied by a flash of light. This could indicate laser cooling. The higher the wavelength the more energy it radiates away. Something in the low ultraviolet would send off more energy and be invisible to humans without being energetic enough to be harmful ionizing radiation. This meshes well with the explanation of their lightning powers.
But, again, this is a lot of energy. For example, freezing 500 g of room temperature water means moving 1 MJ of energy or about 250 grams of TNT. That's quite the flash of light!
Fire
This one's tough. You didn't say "hot", you said "fire". Fire is combustion, an exothermic chemical reaction of matter with oxygen. If they can make fire in air this means they're somehow creating matter for oxygen to combust with.
You can do this from pure energy, but that requires an outrageous amount: $energy = mass \times speed of light^2$. 1 gram of matter contains about 9e13 Joules or 25 Gigawatt-hours or 15 kilotons of TNT so they're probably not doing that. Or are they? Read on.
They could be altering the atomic composition of the air itself, for example, fusing and splitting the abundant nitrogen in the air to form carbon and hydrogen and turning it into combustible methane. But this also requires a lot of energy and implies the sort of extremely fine manipulation you said they do not have.
Lightning
This is probably the simplest to explain from a pure physics perspective. They're doing the same thing lightning does, ionizing air such that it conducts electricity. It happens with lightning because a charge differential builds up between the air and the ground which eventually ionizes the air forming a circuit that charge can travel down.
If you just want lighting to shoot all over, you're acting like a Tesla Coil which ionizes the air around it.
If you want to "shoot" lightning anywhere you like you have to create this ionized channel yourself. We can do this with lasers: fire a laser to turn the air into conductive plasma, then an instant later send your electricity down this channel: an electrolaser.
$e = mc^2$
The fairly simply explanation of where they're getting and where they're putting all this energy is that they're fueling it using the mass-energy equivalence of their own bodies. The caster is converting their own mass into energy and somehow, and this is a big somehow, channeling it. One gram of caster is 25 Gigawatt-hours of energy. A lot. Plenty to power lightning. And an easy way to diet.
For cold, run this in reverse. They're taking energy from the object and converting it into their own mass. The 1 MJ necessary to freeze a glass of water is just $10^{-11}$ kg.
Fire would be very draining because it requires actually creating matter of equal mass that you're losing. Methane burns at about $55.2 MJ/kg$ so might be suitable for the effects you've described, but it would be the most draining of all their powers.
Action at a distance
All action at a distance can be explained by some sort of quantum field that pervades all of spacetime (while gravity is the curvature of spacetime itself). For example, the force you feel between two magnets is conveyed by the electromagnetic field.
Physicists are very sure they have all the fundamental elements of what makes up everyday life. It's called "core theory" and it's an equation that fits on a t-shirt. This covers quantum mechanics, spacetime, gravity, matter, the Higgs field (ie. mass), and the nuclear forces. They know this isn't complete for the whole cosmos, dark matter and dark energy and quantum gravity remain open problems, but for everything we'd experience here on Earth they're very sure they've got everything.
But there's no field to explain what your people are doing. How are they causing energy transfer away from matter at a distance? How are they causing combustion to occur at a distance? This would raise enormous fundamental questions and require tearing physics down to find out what went wrong and how did they miss it in all their experiments? Theoretical physicists would be ecstatic!
Biology
Since these powers are inherited, there's something special about their bodies to allow them to do it. Even if it requires "magic" words or gestures, what is it about their biology that allows only them to do it? What special organ do they have?
2) What might they do with the actual underlying power in a near-future peacetime world, assuming that the control in question isn't fine-tuned enough to affect anything microscopic-scale?
What's with the daggers? Who's carrying a dagger in the near-future? Why would you need a dagger when you can shoot lightning bolts and fire?
Fire effects: Up to a 10-m radius circle of flame that only lasts briefly; the smallest effect we see is a 10 cm sphere that sticks around long enough to use as a fidget toy.
Nobody wants to be lit on fire.
Even if they can't really harm them, they could scare the crap out of a group of assailants with a ring of flame providing a distraction.
They could like a small fire on an attacker's clothes which they will immediately move to put out.
They could heat an object someone is holding causing them to drop it.
Cold effects: 10-m radius circle of cold at something between liquid nitrogen and frozen water temperatures, lasting about a minute
At that liquid nitrogen temperature things become very brittle. It would be useful to break locks, fences... anything obstructing them. They could cool and then shatter their opponent's weapons.
They can put out fires.
Lightning effects: 10-m radius circle of painful but not instantly deadly small "lightning"; 5-m distance fairly serious lightning bolt; most confusingly, a several-minute short-range lightning/sparking effect on a dagger.
They could emulate a stun gun to paralyze foes. Add an electric shock to their punches. Shock opponents while grappling.
With fine control they could stop somebody's heart, or start it again.
They could short out electronics. They could produce strong electric and magnetic fields to jam electronic transmissions: wifi, radio, cell phones, wireless security cameras...
