How sinewave pwm inverter filter output?
Yes the LC filter will do the job. Use a choke of 1mH or more, or 2 x 0.5mH, and 1uf or 2 x 2.2uF.
The commercial inverter has the first stage push pull inverter from 12V to 400V, high frequency - small transformer. Then the second stage is the H bridge, no transformer, output filter. It doesn't mean that your circuit is bad, it is just more heavy, but with only one transistor stage so it can even outperform the commercial in terms of efficiency.
Yes of course. You can add a small transformer on output 220/12 for feedback. Let suppose you have a look up table: sine -> PMW, then you can simply multiply the PWM duty ratio proportionally with voltage drop on the output.  


Link to italian site for inverter DIY