I am a bit confused about the translation of せかいの中の日本、japan in the world, if particle　の is equivalent to ~'s.

Question

I am a bit confused about the translation of せかいの中の日本、japan in the world, if particle　の is  equivalent to ~'s.

Answer

One of the functions of "の" is "~'s(possession)," but that's not the all.  
You can break down せかいの中の日本 into two parts(1 noun phrase and 1 noun) connected by の: 
 「せかいの中」の「日本」. This の(AのB) means "B located at A."   
Again, せかいの中 is broken down into two nouns: 
「せかい」の「なか」. This の（CのD）means "D of C".
So the whole phrase becomes "Japan, located inside of the world".  To make more sense in English, I'd translate it like "Japan amongst the international society" .

Answer

せかいの中 = Connecting the two nouns (noun + の + noun) 
The second の means "of"
So the sentence actually means "Inside the world of Japan".

Answer

It really means just "Japan". And yes, particle "no" means "...'s".