I'm a native English speaker and to me the phrase: "I want to know what's your name" Sounds just as natural. If anything, I'd actually use this over "I want to know what your name is". This is primarily because what you proposed in your examples were questions, whereas these phrases (and the phrase in question, as mentioned by the person above) are not exactly questions. They are rather: statements. Demands. "I want to know..." - demand "Do you know..." - question
the problem is "what" works both as a n interrogative pronoun and a relative pronoun.
It is an interrogative pronoun in direct questions, involving the use of the auxiliary-subject(-verb) construction, but it would here need a change in punctuation : I want to know : "What is your name?"
It is here a relative pronoun involving indirect question, hence the non-inverted construction : I want to know what your name is.
Hope This Helps
I asked a Norwegian, and he said that 'vite' is 'to know', as the example above "jeg vil vite hva du heter" (I want to know what your name is) or "Det er ikke lett å vite" (It isn't easy to know)
And 'vet' is simply 'know' or 'knows' as in 'Jeg vet ditt navn' (I know your name) or 'Hun vet at katten er svart' (She knows that the cat is black).