Voltages are relative, as long as the triac has sufficient gate to A1 voltage it will fire, no matter what the absolute voltages above ground are.
Yes, as long as the output voltages at P1, P2 and P3 are high enough it should work but beware that those output voltages are relative to VSS and the three triacs may not share a common connection with VSS.
Any MCU can detect zero crossing with a suitable interface. Zero crossing point is detected by the absence of voltage, again all you need to ensure is the detector output is relative to VSS.
I'm not sure what your application does but firing in all quadrants would be difficult even with the original plan, opto-isolators make it safer and easier to implement.
Brian.