Part of the problem here is terminology, there are 3 terms: Earth, Ground and 0V that are used interchangeably as if they mean the same thing. Not only do they not mean the same thing, I get the impression, not verified, that earth and ground do not mean the same thing in different parts of the English speaking world. What they mean in parts of the world where English is not the primary language I do not know.
Those terms are used interchangeably because they are often connected together so one connection ends up doing different but related things. I am hesitating to explain more because when I wrote the common ground tutorial I decided it was going to be difficult to get it right and it was well beyond the scope of the problem I was trying to address.
Your lack of electrical earth from the supply is a safety issue not an issue for the functioning of the circuit although it might be an issue for the continued functioning of your body given the safety implications. You are already getting shocks, be careful!
EDIT
I started typing the above before @DVDdoug posted reply #5 , but his reply was there by the time I'd finished. He explanation of the US system adds to my reluctance to define earth, ground and 0V. The US system is different to the UK system, although there are some similarities and both are trying to achieve safety.