Is there any difference among "sound native", "sound nativelike" and "sound natural" when it comes to the following sentence? "He speaks English fluently and sounds native / nativelike / natural."
Your example sentence
He speaks English fluently and sounds native / nativelike / natural.
suffers from the problem of saying the same think twice, at least approximately.
He sounds like a native
would be natural as a stand alone sentence.
I personally would avoid the use of native too widely as in my childhood it was used to describe the people of our colonies in Africa and conveyed the sense that the speaker thought of them as rather inferior, a common view at the time. I suspect most younger speakers would not have that memory so it is probably acceptable these days. It is fine if you mean someone was born somewhere, as in he is a native of Wales.
"sounds native" is of a person.
"sounds natural" is of a sentence or of a person's speech.
"He sounds natural" doesn't really... have a lot of meaning.
"That sentence sounds native" also sounds odd.
You can say "that sentence sounds natural" or "that person sounds native" and they make perfect sense, but no, overall, the adjectives do not have the same meanings.