Noah's ark

[Director Michael]

Please explain further the laws of appropriate partners to me. An objection someone who does not believe in there being a real Flood and Ark is that there were only four male/female couples on the ark, and after they left the ark, their children would have had to find mates from among their own cousins on their paternal side. Does the Noahide Code only restrict relations from among one's mother's relatives but permit the father's side?

That is correct in regard to the category of restrictions on relations between blood relatives. See our web page
https://asknoah.org/7-commandments/forbidden-relations

Also, a Jewish friend says she has been taught that there were other floods before the one Noah & his family survived, and that the Flood in Noah's days only covered 1/3rd of the land, so there could have been other survivors.

Your friend is just a bit confused about this. The Midrash says that there was a limited flood (covering 1/3 of the world) as a warning to mankind in the days of Enosh, before the world-wide flood in the days of Noah. See Rashi's explanation on Genesis 6:4 (in which he cites this Midrash) -

"The Nephilim were on the earth in those days, and also afterward..." Although they had seen the destruction of the generation of Enosh, when the ocean rose up and inundated a third of the world, the generation of the Flood [in Noah's time] did not humble themselves to learn from them.