Can I change "I want him to go to the zoo" into "I want to go to the zoo, his going"?
I saw lyrics in National Anthem by Lana Del Rey: "He said to be cool, but I'm already coolest," not "he said me to be cool."
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"I want to verb" means you're talking about yourself, and a thing you want to do in the future. If you're talking about someone else, you have to say "I want X to verb" (X is the person or thing doing the verb).
In your Lana Del Rey example, the person who spoke to her said "be cool". That's direct reported speech, and in the song it's indirect reported speech: "he said to be cool". It's like someone saying "go home!", and you report it as "she told me to go home".
The expression "I want to go to the zoo, his going" is ungrammatical. The part before the comma is correct. There is no analogy between this and the correct "He said to be cool, but I'm already coolest".
The lyrics contain two coordinated clauses "He said to be cool". The verb say (unlike the verb "tell") does not take an object "me". The expression "He said me to be cool" is ungrammatical. You can't put "me" in that sentence. You could say "He told me to be cool". The second part has a different subject "I'm already coolest". There is some word play here, the meaning of "cool" in the first sentence may be "not angry" but in the second it is "stylish and popular".
But "want" can have a pronoun: "I want him to go to the zoo" means that "I hope that he will go to the zoo, even if I don't" It doesn't mean that "I want to go to the zoo with him".