Is there any difference between "live life" and "live a life" in daily conversations?
"live life" and "live a life"
In "live life" life is uncountable and refers to that which is experienced whilst living, i.e. "life in general".
"Live a life" is, as Edwin Ashworth points out, wrong because it is incomplete - it requires a complement, i.e. "Live a life of happiness/worthy of your calling/that people will remember, etc."
In "Live a life of happiness", "life is countable as it is one life from many possible types of life, and "a" = an example of a life of happiness.