Do nen examiners also have to teach them Nen? Gon and Kurapika's examiner also taught them nen. Is every examiner also a 
 teacher, or the hunters have to learn nen by themselves?
I belive you're refering to the "Secret Exam" when you talk about "examiners". I'm not sure this is entirely accurate.

There doesn't seem to be a very strict structure to the Secret Exam, at least compared to the Public Exam. As such, there doesn't really seem to be "examiners" as such. The hunters just seem to collectively keep tabs on the passing participants to see if they figure things out on their own. Some actually teach these people Nen, but none seem exactly obligated to do so.

So, in short, there propably isn't the sort of requirement you mention, because there aren't really any "examiners" at this point. All teachers we see seem to be giving their lessions more or less out of their free will, rather than any sort of obligation. Of coures, almost everybody capable of teaching Nen is already a hunter (with the some exceptions like the Phantom Troupe, who know Nen but aren't necessarily hunters), so they would be qualified to pass those they teach, when they seem ready. But remember that there are people who already know Nen, who pass the Secret Exam the moment they pass the Public Exam, without any sort of fanfare.

Basicly and simplified to the extreme, I'm proposing that in stead of any sort of examiners, just about any hunter can just contact the Hunter Association and say "Yeah, they know Nen now. They're real hunters alright." Of course, the exact details of the Secret Exam are rather vague, so the answer is rather speculative.

As a side note, I think it would be theoretically possible for someone to pass the Public Exam, learn Nen from a non-hunter and then pass the Secret Exam when other hunters learn about this. But there isn't really an example of such thing ever happening.