Why output signal is not symmetrical? What to do?
Your problem is that the base-emitter junction in the common-emitter amplifier does not have a stable voltage. Hence the gain is variable dependent on where you are on your output wave.

Your quiescent emitter current is around \$1.5\ \mathrm{mA}\$. Hence \$r_\mathrm{e} \approx 16.6\ \mathrm{\Omega}\$. If the output swings by half a volt, the collector current swings by

$$\frac{0.5}{5000} = 0.1\ \mathrm{mA}.$$

The current swings in both directions, hence \$i \in [1.4, 1.6]\ \mathrm{mA}\$. Thus \$r_\mathrm{e} \in [15.6, 17.9]\$. Therefore the gain is in the range \$A \in -[320, 279]\$.

There are various ways to fix this problem: decrease the emitter capacitor so its impedance is larger than the dynamic resistance; increase the biasing current so that the dynamic resistance is smaller; put a resistor in series with the emitter capacitor such that the voltage drop across said resistor is larger than the base-emitter voltage.