Question: Lichvod Ha'Rav: Is there any 'mitzvah' to try to influence a non-jew, son of a jewish-father, to convert?
Certainly as a general rule we do not try to persuade non-Jews to convert, and having a Jewish father would not create an exception. The only possibility I can imagine is if drawing the son closer to Judaism would make it more likely to induce the father to be chozer b'teshuva. However this seems a little too far-fetched to me.
Question: Lichvod Ha'Rav: Would it be me'akev for a would-be woman convert to accept covering her hair ? (Since not everybody does.., or other reason)
This is a sensitive and complicated issue which must be asked to a Rav proficient in the Halachos of geirus and personally familiar with the person.
The general rule is that a conversion is invalid even retroactively if the convert knowingly refused to accept at the time of their conversion a single Halacha or even minhag. However if they accepted everything they knew to be obligatory but were inaccurately misinformed that covering the hair is merely a stringency, the conversion would be valid and they would have the same status as any Jew.