Are all 0W-20 and 5W-20 ils fully synthetic?

No.               

Nope.  I've purchased conventional 5W20 engine oil before.

No - only the ones that say they are fully synthetic.

No.   They can make regular oil as thin as you want.  

NOT ANSWERABLE. Those numbers indicate viscosity. That says nothing about being synthetic or not.

All 0W-20 oils are either full synthetic or synthetic blend. 5W-20 can be convention or synthetic.