"Curiosity", "hatred", and (I think) nouns for other emotional states are not countable. However, these emotions can be figuratively associated with things, and those are countable. For example:
"In his youth, he had many loves."
"The hate he had for his oppressors was nothing compared with the hate he had for those who collaborated with them."
"Her children were foremost of the many joys she had in her long life."
This is not really counting emotions, but more counting the things associated with those emotions -- not "many loves" but actually "many people that he loved"; not "two hates", but "two things that he hated"; not "many joys" but "many things she felt joy for".
I've not heard this used with "curiosity" (as an emotional state) but it's possible. Your second example "hatreds against his rival" isn't wrong but still doesn't make sense because how could someone be hated more than once?
All that being said, it is possible to see plural emotions used in slang for humorous effect. For example, "I haz sads," or, "Full of happies!" or, "On Mondays, the blahs always win."