Learn English – “Which part don’t you understand?” or “Which part you don’t understand ?”

I: a subject

don't: a contraction of "do (an auxiliary verb) not (an adverb to form the negative)" whose function is negate the sentence (form the negative)

know: a main verb

when (an interrogative adverb) to do (to-infinitive) it (a pronoun): an object of the transitive verb "know".

"When to do it" is a wh-clause which is short for "when I should do it". I would not call it a noun phrase which has a different definition as the link indicates.

As the subject of the sentence "I" is the same subject of the "when clause (when I should do it)", "when I should do it" could be shortened to "when to do it".

You can visit the above link and read more about how the "wh-clause" works.

Want is a transitive verb in English and does not make sense if it is used without a direct object.

Both of your examples have direct objects.

I want to be famous.

Direct object: "to be famous"

"To be famous" is an infinitive phrase and infinitive phrases are a sub-category of noun phrases. Therefore, the entire phrase can be thought of as a noun. As far as English is concerned, in the following sentences,

I want to be famous

I want a cat

"cat" and "to be famous" serve the same function.

In your second example, "to be famous" is still the direct object. "You" is something called the indirect object.