What is the difference between 中国歌, 中国之歌 and 中国的歌? When do we use each of the following cases? Is there even a difference between them?
中国歌 means "China song" or "Chinese song"; 中国 (China) is an adjectival noun-- the song is either about China or it is in the Chinese language. 
中国之歌 means "song of China"; 之 is a (relative or true) possessive determiner-- The song is either about China or China actually possesses the song. 
中国的歌 means "China's song"; 的 is a (true) possessive determiner-- China actually possesses the song. 


Difference between true possessive and relative possessive:

之 in 将死之人 is a relative possessive determiner. 将死 is not a noun, therefore it can't possess the object 人. 之 only indicates 人 belong to the state of 将死.

If 中国 in 中国之歌 is the subject noun, then 之 would indicate the song belong to China--it is China's song.

If 中国 in 中国之歌 is an adjectival noun, then 之 would indicate the song is about China.