Electronic – What operation could cause the inductor or the capacitor to explode

To answer this properly, you should know the properties of a capacitor and an inductor.

Inductors are one of the primary components required by a switching regulator. A capacitor and an inductor are similar in the way that a capacitor resists a change of a voltage and an inductor resists a change in current. The "strength" of their resistance depends on their value

Capacitors are widely used to clean up a power supply line, i.e. remove noise or ripple at (higher) frequencies. Inductors are used in switching power supplies where a relatively constant current is passed through an inductor. A switching power supply works in that a switch is opened and closed very quickly. When the switch is closed, the inductor is 'charged'. When the switch is open, the energy is drawn from the inductor into the load. Usually such a power supply is being decoupled with a capacitor to create a stable power supply line.

An inductor is required to make this principle work. If you know a resistor that has an equal resistance for all frequencies of signal, you should view a capacitor as a resistor that will be infinite for DC (0Hz) and 0 for high frequencies. An inductor will be the opposite: it's resistance will be 0 at 0Hz, and infinite at high frequencies. However we don't call this resistance (that's only used for a pure resistor!) but impedance.

A PC motherboard or graphics card is basically not much else than this. They have their main chips and the routing between them, and most other components are power supply or a little bit of interfacing between chips or connectors.

It sounds like your understanding of the theory is missing an important point: Each turn contributes to the EMF. There's a big difference between one turn of large wire, and 1000 adjacent turns. For a large number of coils close together, Faraday's law of induction can be restated as

$$ \mathcal{E}= -N \frac{d\phi}{dt} $$

where \$N\$ is the number of turns, and \$\frac{d\phi}{dt}\$ is the rate of change of flux. For simplicity, this is often developed using one loop.

The wires used to wind the generator coils may look like plain copper wires, but they are actually coated in a thin layer of enamel, which electrically insulates them from adjacent wires.

I believe that the mesh you show has un-enameled wire (Test for continuity between crossing wires to be sure) which connects at every crossing. Using this to wind a generator would result in one turn, giving you an EMF \$N\$ times smaller than what you'd get with an equivalent number of turns of magnet wire.

If the mesh does have enameled wire, and you can make the connections such that each wire contributes to a loop, and you can bend the mesh such that it occupies little more space than the equivalent number of turns of magnet wire, I see no reason why it would function any more poorly than the magnet wire version. That said, it's an awful lot of work for little-to-no gain. Get some magnet wire, find a university or business with a coil winding machine (or buy one/make one with an electric drill and footswitch), and do it right.