Not understanding aspects of the LM386 amp
C3 and R1 is called a zobel network - go google go.
C2 stops the natural DC offset from the amplifier being shorted to ground by the speakers dc resistance.
A input DC block capacitor ought to be used. If in doubt use it. This doesn't mean C2 isn't needed. Read the data sheet.
A power of 0.25 watts being dissipated by an 8 ohm resistor is an RMS voltage of 1.4142 volts, yes.
No, C3 and R1 is a zobel network used to linearize the impedance of the speaker and make it look more like a fixed value resistor across the audio spectrum. Nobody hardly ever uses them these days but some amps will be unstable without it.