When and why we use double-genitive with respect to the word 'of'?

I wanted to know why and when to use double-genitive. So for example why can't we use 'I am a fan of YOU' instead of 'I am fan of YOURS'. This is so as using the word 'of' itself meant that the noun mentioned first (before 'of') already belongs to the second noun (after 'of'). For example, in 'I am a fan of YOU' , the word 'of', to me, already suggests that 'I am your fan' without using 'YOURS'. It is like if we use 'fan of YOURS', it meant like 'I am your fan that belongs to you'.

So why is it necessary to use the double-genitive here?

A similar question has been asked: Why use apostrophe-s to denote possession when using 'of'

But one of the answers said that it is optional to put apostrophe for the example given in the question except to clarify the meaning of the sentence. (This is a picture of Bill/Bill's) So, what are some examples that are compulsory to use the double-genitive?

For examples, is it:

1) 'fan of YOURS' or 'fan of YOU'?

2) 'The death of HIS is...' or 'The death of HIM is...'

3) 'friend of HER' or 'friend of HERS'

4) 'queen of England' or 'queen of England's'

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On a side note, we say:

5) 'meaning of the WORD' instead of 'meaning of the WORD'S'

6) 'son of my FRIEND' instead of 'son of my FRIEND'S'

7) 'plays of Shakespeare' instead of 'plays of Shakespeare's'

8) 'city of Rome' instead of 'city of Rome's'

Overall, my question is why is it that we sometimes uses double-genitive and sometimes don't? Is it because of whether the noun before 'of' is an object or a person except in the case of 6)? If not, when do we use double-genitive and why? (in the event that the meaning of the sentence doesn't change regardless of it being double-genitive or not)