I've seen that [MAX98357A] accepts 2.5-5.5 V as input and I'm wondering if it has any impact on the output. Do I get louder sound if I put 5 V in? Do I get noisier sound?
The datasheet claims "3.2W Output Power into 4Ω at 5V." If the voltage is reduced, the power it can deliver will be reduced as well, but probably more than expected. Claims 0.4W at 2.5V. There's a chart of this on page 11. So that's the main reason for using 5V if possible.
The charts on page 9 show generally, that noise is very low (-70dB or so) for all but the highest power outputs. So, it should sound fine at any low volume, but crank it to max, and it'll start sounding bad.
By "bypassing" they mean to connect a capacitor in parallel to the input source, right? When they require both 0.1uF and 10uF caps, I guess they are both in parallel, like in this schematic?
If bypassing is required, why isn't it a part of the MAX98257A board itself? Should I adjust the values to my specific use-case? If so - how?
It isn't part of the board itself because they assume you're not going to power it from a long (and resistive) cable. It's also cheaper to make the board with less parts on it (offloading design caveats to you.)
If you are going to power the board from a long cable, then add the 0.1µF at the board. And if powering it from a mile of cable, even more capacitance will be needed.
Generally this is analyzed using an oscilloscope. The oscilloscope is setup to measure the voltage changes at the board, and capacitance is added to reduce these voltage changes. The amount of capacitance depends on how good (low-impedance) the power supply is, and how lossy the wiring is. Math could be used, but a lot of variables are involved (calculating resistance from wire diameter/length, etc.) It's easier to just measure it. The datasheet says that a 0.1µF may be needed and that should cover most use-cases.
What exactly do I protect here? Noise coming from the power supply or noise that the amplifier generates (on the speaker cables)?
Think about what would happen if instead of being directly powered from the power supply, it was instead powered through two 10 Ohm resistors. When the audio volume would peak, the voltage would sag badly... likely enough to cause undesired operation.
Well very fast changes in this audio ("transients") can put a tremendous strain on any power supply, such that even 0.1 Ohm of resistance (i.e. a long wire) will cause the voltage to briefly dip, leading to undesired operation. The 0.1µF cap next to the board is to briefly supply power during these events, so that the volts doesn't decrease too much. You should see a noticeable reduction in variation when the cap is added.
Should I also put capacitors on the speaker cables to get rid of "ticks and some static noises"?
In general, it is possible to put a capacitor (such as 10µF) in series with a speaker. This is because capacitors couple AC and block DC, so have the effect of eliminating any DC offset present in the output. But a capacitor should not be placed across a speaker, as this will be shorting the AC signal out.