The sefer Hashavas Aveida k'Halalcha quotes HaRav Eliyashiv shlita as saying that non-Kosher food is not automatically exempt from the obligation to return a lost object. However if there is reason to suspect the person may consume the non-kosher food, it is better to sell the item and return the money. However there are other issues involved. If the lost item can be assumed to belong to a non-Jew, for example it was found in a city that is primarily not Jewish, then there is no obligation to return it. Additionally, if the city is majority non-Jews there is no need to return even a Jew's lost object, because he assumes that it was found by a non-Jew who will not bother trying to return it so he makes it hefker immediately.

The Rema mentions a minhag not to go to the mikva Friday night (and Poskim add also motzai Shabbos) if she is not going to the mikva on the earliest possible date. Often women after childbirth postpone going to the mikva until they are fully healed, even if they stopped bleeding earlier. In any event, even if the woman is going to the mikva late, the Rema does not endorse this minhag and says one should go anyways, unless the city or the individual has a specific well known minhag to be machmir. The Poskim write that if one must inquire whether or not this is the minhag, it is not considered "well-known" and one should not be machmir.

So there is some basis to what you heard, but it is not the accepted practice.

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.

The minhag to recite Kiddush in shul Friday night originated at a time when guests frequently ate and slept in shul, then it would have been necessary for an adult to make Kiddush to fulfil their obligation. However for many centuries this has not been the case, and in fact the Shulchan Aruch OC 269 recommends not saying Kiddush in shul at all since Kiddush may only be recited where a meal is being eaten. While the Mishna Berura there 4 writes that the long-standing minhag to recite Kiddush should not be abandoned, I see no problem allowing a minor to recite it.