Q) "Hate and love themselves are relative?"

A) No, they are complex, mixed...adulterated.

Q) "Are they not merely emotions?".

A) To say they are emotions is to state the obvious, but emotions are hardly 'mere' from the human perspective [i.e., the only perspective we have]

Q) "To the philosopher they exist in much purer forms than what most people think them to be".

A) This is Platonism. The theory of Forms posited 'pure' emotions etc. Nietzsche destroys this kind of metaphysics and views the emotions ['affects'] from the basis of physiology/psychology.

Q) "Of course, hate and love are always recognized as Christian sentiments: that is that they are directed toward another".

A) No, only in the sense that Christianity inherited the Platonic Forms;"Christianity is Platonism for the People" [Nietzsche BGE Preface]

"One loves ultimately one's desires, not the thing desired." [N.]
"What is done out of love always takes place beyond good and evil." [ib.]
"One does not hate as long as one disesteems, but only when one esteems equal or superior." [ib.]

Q) "This is why much higher men hate very few. In such an age as we are in, it is no wonder they hate no one".

A) It is not the hate of the 'higher men' that we are worried about! In your egalitarian future there will be more and more hate confused with sentimental love; so much so, that the politically correct will put to death thinkers while smiling that they are doing it out of love!

It follows that man must become more wicked and terrifying if you want to increase love in the world.
But you will never achieve the ideal of 'pure love', that 'world without war' so beloved of sentimental liberals.