We need more information to comment on this in detail.
Signal frequency, switching frequency and the full circuit with component values, not just a block diagram.
>-40dBc 2nd harmonic is observed at the output.
Class D amps produce some harmonic distortion. That's normal. -40dB is acceptable for some audio applications. If you want very low distortion then don't use a class D amp.
>overshot (>VDD=3V or <VSS=0V) are observed,
The input of the filter will be reactive. The inductor and the capacitor will be series resonant at some frequency. Ringing is expected behaviour.
>waveform above VDD is not symmetric to that below VSS.
Is the circuit of the H-1 block symetrical?
We need to see it to comment.