Is the grammar of this sentence good?

Yes, that is OK.

What would be the function of "somewhere you can be alone"? Is it A. "somewhere you can be alone" = adverbial clause or B. "somewhere" = noun; + "(that/where) you can be alone" = relative clause

The latter, it seems to me.

Isn't it that after "take you" there should be an adverb function? I'm really sorry but an American teacher proofread what I wrote and he says the sentence (Do you want me to take you somewhere you can be alone?) is wrong. But I've heard the clause "somewhere I belong" many times. So, I don't see why "somewhere you can be alone" should be wrong. I'm just following the syntax "somewhere + S

There is nothing wrong with the sentence. 'Take you' can easily be followed by an object: Take you home, take you somewhere.

Somewhere:noun: an indefinite or unknown location

Are you sure 'home' and 'somewhere' are not adverbs there? Because they both answer the interrogative question "Where do you want me to take you?" and not "What do you want me to take you?" "What do you want me to bring you?" may be fine, but that's a totally different sense. It is answerable by "I want you to bring me some drinks (or my bag)".

I am happy in considering them locations. My home at least is, and when I ask my pal to drive me home, I am thinking of it, not a modifier. No doubt you can find conflicting opinions on the web, though, so feel free. I have done.