Q.What are the functions of input and output coupling capacitors?

A capacitor offers very low reactance to AC but blocks DC completely, and amplifiers exploit this in two distinct roles.

Coupling capacitors sit at the input and output (and between stages). The input coupling capacitor passes the weak AC signal to the base while preventing the source from disturbing the carefully set DC bias. The output coupling capacitor delivers the amplified AC to the load or next stage while blocking the DC collector voltage from reaching it. In short, coupling capacitors keep the signal path continuous for AC while isolating the DC operating point of each stage.

The bypass capacitor $C_E$ is connected across the emitter resistor $R_E$ in a CE amplifier (or $C_S$ across $R_S$ in a JFET amplifier). The emitter resistor is needed for DC bias stability, but if the AC signal also flowed through it, the resulting AC negative feedback would sharply cut the voltage gain. The bypass capacitor provides a low-reactance path that shunts the AC emitter current around $R_E$, effectively grounding the emitter for AC. Thus $R_E$ still stabilises the DC operating point while the capacitor prevents any loss of AC gain — the capacitor behaves as an open circuit for DC and a short circuit for AC.