The City of London is actually a small area inside the large city most people refer to as London. Most of the surrounding city is actually Westminster. To refer to the city, then use the City of London. To refer to the metropolitan area, just call it the London area or simply London and clarify if it is needed.

To avoid confusion, do not say "the city of London" unless you are referring to the City of London (note capital C), which is a separate city within London known by that name. See this Wikipedia page to understand what is meant by the City of London.

I would say both are right. However, London is extremely common and "the city of" is pretty much implied. If you are speaking to a group and there is a chance they don't know what London is, say "city of London". But any other time it's okay to just say "London".

They're both correct. Use either one. You might want to distinguish between the city of London and a sports team from London, or a person named London, or a movie named London.