Emitter resistor in common-emitter amplifier
No, Re does effect voltage gain.  To first approximation, the voltage gain of \$V_o/V_i\$ is \$-R_c/R_e\$ at DC.  Due to \$C_1\$, the gain increases with frequency near and above the rolloff frequency of the filter formed by \$R_e\$ and \$C_1\$.

To chose \$R_e\$, make it \$R_c/G\$, with \$G\$ being the desired voltage gain of the circuit.  G should be a few times less than the current gain of the transistor, else the gain won't be predictable and will have significant dependence on the transistor gain.  Being dependent on the the transistor gain is not good for repeatability since transistor gains vary quite a lot even within the same model.  That is why datasheets usually only tell you the minimum guaranteed gain at a few operating points.  The actual gain of any one part can be significantly more, like 5-10x the minimum.  When predictable gain matters, you design the circuit so that any transistor with a gain from the guaranteed minimum to infinity will still let the circuit operate correctly.  A emitter resistor setting a smaller gain is one way of achieving this.