Roles of the coupling and bypass capacitors in a CE transistor
The role of C3:
It is the main purpose of C3 to restrict the negative feedback effect (caused by Re) to DC and very low frequencies (below the desired operating frequencies).
This feedback effect (for DC) is very inportant because it makes the DC operating point less sensitive to parts tolerances and variations of the transistors B-Value (B=Ic/Ib).
As another effect, negative feedback reduces the gain value - and if somebody does not want such a reduction, the feedback effect must be cancelled for operating frequencies (bypassing Re with a capacitor C3).