In thinking about the various relationships you describe I believe that the various forms of love depend on the nature of the attachments we have and what we project on to various people or animals we love. Some of attachments may be about bonds, for example we may make certain friendships because we can identify with the particular individual, and their can be a unified experience of sharing. In a sense, perhaps all the basics of love begin with meaningful connections.
Of course, part of the criteria for love involves sexuality. This is obvious in the case of a partner, but possibly, to some extent, this is true of all relationships with other people (but not with one's dog hopefully.) Of course this does go back to the Freudian tradition but it does not have to be purely about the Oedipus complex, although it may be part of the picture.
But going away from that concept, we could say that all relationships are about self expression, and this has a sexual component. This would include same sex relationships too. Of course, many people would claim that they are completely heterosexual.Perhaps they are, in the sense that they would not have gay sex but there may be some homoerotic feelings at the basis of friendships, even though these might not be named by themselves in such terms.
As far as hate goes, perhaps the various forms of hate depend on why someone hates another and the degree of hate. For example, someone may feel a certain amount of hate towards someone who is more successful, but this may just be passing thoughts. On the other hand, someone may hate someone incredibly because they project onto others aspects of themselves they hate or fear. This may be what happens in the case of people becoming bullies,and in this example the victim may start to hate the bully. But the intensity of feelings and duration are probably significant to hate becoming a real problem.
But I do think that it is a paradox as you suggest. In particular, there can be complex ambivalence. For example, a person might have ambivalence to an aging parent. This could be a combination of affection, past sentimental memories as well as gratefulness in favour of love. But this could be combined with hate for possible hurts from the past, difficulty with watching the parent decline and feeling trapped by the demands of the care needs of the aging parent. So, ambivalence between love and hate can be based on the whole spectrum of emotionally charged experiences.
The whole depths of love and hate are so complicated. Perhaps the best we can do is try to be conscious of the emotions. Personally, I try to aim to seek loving thoughts rather than dwell on hateful ones. But not to the point of denying them entirely. I think that mindfulness is helpful here. But of course, we need a bit of release, which can include talking to others which can be helpful, but this varies depending on their own mindset. And, for dealing with love and hate and it's difficulties that is where a bit of music comes in. I definitely have my heavy metal days.....