this is ok: the Queen of England's reign
but how to make a plural? the Queens of England's reigns?
or: his father-in-law's car
I also found on http://view.byu.edu/ such phrases:
his sons heirs; shouldn't it be 'his sons' heirs'?
his grandparents memories; or his grand parents' memories?
These are separate questions:
1) The 'Queen of England' in your first sentence is a composite noun - it is an atomic grammatical unit. Hence, we say "The Queen of England's reign" by the same rules as "The Queen's reign."
2) is more complicated. It depends upon whether common usage indicates that "Queen of England" is a useful CLASS (or noun), a word describing a type of person, place or thing. If it were, we would say "the Queen of Englands' rule". However, this is silly - there is only one 'Queen of England' at any time, and it is therefore a poor excuse for a noun. We would also be obliged to write 'Queen-of-England' in order to indicate that we are referring to a composite noun.
This question is another problem for English, because the rules of grammar here are inconsistent with usage. Take the most well-known case: 'Trade Union' - an organization negotiating with management on behalf of workers. Generations of English children have been drilled with the idea that the plural is 'Trades Union' because English teachers convinced them that Trade-Union is not a composite noun, but two entirely unrelated nouns. The gathering of all these Trade-Unions is thus religiously called the Trades Union Congress, even though no native speaker would dream of saying "I have been a member of two Trades Union", nor would they know why they should.
But back to the point... it should be:
2) "The Queens' of England reigns". The reigns belong to the Queens, and 'of England' is just an adjective here. It is such a mess that a native speaker would never use the Saxon genitive here - they would always say "The reigns of the Queens of England."
I hope this is sufficient explanation to cover your other examples, so I will just give answers:
3) "His father-in-law's car." and "His father-in-laws' car."
It's an interesting question. I'm more inclined to look favourably on such constructions as
1. The Queens of England's reigns.
These for instance are idiomatic:
2. The Queen of England's reign.
3. The Duke of Westminster's vast fortune.
The 's ending seems to function as a clitic, in modern English usage (i.e. as a grammatical element that is independent in terms of syntax, but which is attached to the preceding word). In everyday English, it can be attached to some extraordinarily long strings:
4. It's the sister of the girl who works in the bookshop on Saturday next to the cinema's handbag =
4a. It's the handbag of the sister of the girl who works in the bookshop, etc.