Finding AC equivalent circuit of a BJT amplifier
In order to find the operating point for your circuit you have to perform the steps 1-5 given above. The operating point is equivalent to the DC analysis of the circuit. Inductors and capacitors are therefore removed by a short or open circuit, respectively.

For this DC solution the circuit is linearized. Your transistor is replaced by a simple transconductance (voltage-controlled current source). The AC components are of course required for the AC model of the circuit and need to be included again.

To answer your questions:


C1 is left in the circuit because we need the AC model of the circuit
The transistor is now a voltage-controlled current source, with a transconductance gm
Vpi is simple the controlling voltage (the base-emitter voltage) of the transistor
Re is the emitter resistor which is given as 1kOhm