Skin effect is relevant for any change.
If you prefer a transient (time) rather than Fourier (frequency) domain analysis, think of it this way: when a current is applied to a wire, at first the EM field around the wire carries the energy. Over time, the energy "soaks" into the conductor. At first, the conductor prohibits fields inside it, because it is conductive: a conductor shorts out the E field, drawing eddy currents on its surface, which produce a magnetic field which opposes the applied field.
No conductor is perfect*, and the opposing magnetic field is not perfectly opposing. The outermost surface opposes just a small amount of the external field. Each layer down does its job as well, and eventually, the field is stopped. But each layer gives way over time, giving the effect that the applied field soaks in. Eventually, it soaks all the way through, and the full wire cross section ends up carrying a DC current.
By "soak", I mean diffusion: much as smells diffuse through still air, electricity diffuses through conductive materials. Diffusion has a characteristic sqrt(t) or 1/sqrt(f) property. Hence, skin depth is proportional to 1/sqrt(f).
*Except for superconductors, which exhibit skin effect (the outer layer of ~100 nm, depending on material) down to DC. Except for type 2 superconductors, which exhibit hysteresis loss (flux pinning). Though that's a topological change, not skin effect.
This is all very low level and theoretical; what does it imply for a capacitor discharge? Well, if the coil has very little inductance, then the risetime will be fast, and only the outer layers of the coil will absorb the energy.
FYI, merely shorting out a film capacitor can deliver a pulse of fractional megawatts!
It is difficult for a beginner to understand just how sudden, how powerful, a spark is! One might be familiar with different kinds of shocks and surges seen in every day life: clapping hands, the ping of a hammer, the pop of a firecracker, or a gun. All of these events are faster than human experience can understand: they occur in milliseconds or less, while our senses can't resolve much better than 10 or 20 milliseconds. They are so fast, that they seem to happen instantly. They could happen in one microsecond or one millisecond, and we can't tell the difference!
So it is for this reason, why it's hard to appreciate how destructive an electrical spark can be. An electrolytic capacitor can discharge in ten microseconds; a film capacitor in hundreds of nanoseconds; indeed, ESD -- the mere touch of a thin spark from your fingertip -- goes off in just a few nanoseconds, a million times faster than your senses!
Now, that said: is there any hazard to the coil? No. Very doubtful anyway. Copper is very conductive, even on short time scales (microseconds). The skin depth might be on the order of 0.2mm (give or take). A larger wire will not get very hot, even at the surface, even for those microseconds. A smaller wire may get hot, but you need to compute the (energy / heat capacity) to tell how much.
Can it ever be a problem? Absolutely! When fast capacitors are discharged into a small piece of wire, the wire will explode. The explosion happens in less than a microsecond, yet the speed of sound across the wire diameter might be tens of microseconds: the explosion expands faster than the speed of sound, in other words, it's created a shock wave! A shock wave is all it takes to detonate a high explosive, so this mechanism is very popular -- safe and effective -- for igniting them (exploding bridgewire).
In summary: things in electronics can happen a whole lot faster than anything you might be familiar with; but don't worry, as the laws of physics stand firm. There is nothing that cannot be calculated here! And solid materials, like metals, are very robust, just mind their limits.
Tim