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I * V = W
1A of current @ 1V is: 1W
1A of current @ 10V is: 10W

I understand that 1A is 1C-per-second, which is some amount of electrons. Intuitive so far.
How can I get more power if I use the same amount of electrons-per-second flowing through?

Using the water analogy, where voltage is the pressure of the water and current is the amount of water that flows through, using more pressure also gives us more water flow, so they aren't independent variables. Using another water analogy, where voltage is the height of a waterfall, this one seems to work because the water has some time to accelerate and have a bigger impact on the ground (more power), but I can't find where is the equivalent effect in electricity - what causes the acceleration, and what are the velocity * mass here?

What exactly happens when you increase the voltage in a circuit, without changing the number of electrons passing though per second?

Another analogy to voltage is how much the electrons want to move. But if they only want to move (Voltage) but aren't actually moving (Current stays the same), how can this affect something? (power)

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power is the instantaneous product of current times voltage. Increasing the source voltage causes the power delivered to increase.

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One reason that the water analogy is often a bad idea is that people usually don’t understand how hydraulics work either, so you have to explain the how water works in order to get a bad analogy for how electricity works. In your case:

Using the water analogy, where voltage is the pressure of the water and current is the amount of water that flows through, using more pressure also gives us more water flow, so they aren't independent variables.

The pressure and the flow are not strictly dependent on each other. You can easily increase one without increasing the other. In fact, the human body can change blood flow to an organ quite dramatically without changing the pressure much. It does this by dilating the vessels. This increases flow without increasing pressure. Your statement applies to plumbing, but it is not a general fact.

So, since the water analogy is more trouble than it is worth then it is best, IMO, to simply learn circuit theory directly. $P=IV$ could hardly be simpler. Any analogy that you introduce will add needless complication. A bit of dimensional analysis is useful: current is charge/time and voltage is energy/charge so current times voltage is energy/time which is power.

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For resistive situations, higher voltages with the same current requires more resistance. More electrons are scattered by the crystal lattice for the same current. And that scattering is the source of heat.

The power of a lamp of 100 W under 110 V (and 0,9 A) is much bigger than the lamp of a car of 11W under 12V (and the same 0,9 A).

For the water analogy: a narrow hose can deliver the same flow rate than a wider one if the pressure is increased. But the mechanical power (to remove dirt of a floor for example) is bigger.

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Using another water analogy, where voltage is the height of a waterfall, this one seems to work because the water has some time to accelerate and have a bigger impact on the ground (more power), but I can't find where is the equivalent effect in electricity - what causes the acceleration,

Electric field causes the acceleration.

If you want a closer equivalence to the waterfall scenario, where (roughly) nothing prevents the particles from continuing to accelerate as they "fall" down the potential gradient, think about a current in a vacuum tube rather than in a wire.

and what are the velocity * mass here?

Current.

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Using the water analogy, where voltage is the pressure of the water and current is the amount of water that flows through, using more pressure also gives us more water flow,

Only if the restrictions are the same. You can hold the flow rate constant even as the pressure increases if you increase the load as well. Falling water can spin a turbine, but it doesn't have to fall fast. You could have a pipe with the water moving slowly. As long as the turbine is well matched, you can get equivalent energy out.

Using another water analogy, where voltage is the height of a waterfall, this one seems to work because the water has some time to accelerate and have a bigger impact on the ground (more power)

Correct, as long as there is no resistance to the fall. If you instead had it in a pipe, the water would still gain energy as it fell, even though it wasn't accelerating. It's just the energy would be going into the load rather than increasing the KE of the water.

What exactly happens when you increase the voltage in a circuit, without changing the number of electrons passing though per second?

You've had to increase the resistance, or the ability to pull energy from the circuit. In the water analogy this is similar to increasing the load on the turbine while simultaneously increasing the water height. Increased pressure, increased power, same flow rate.

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