I didn’t realize that we were talking about a non-frum Jewish person’s kitchen (I thought we were talking about non-kosher dairy restaurant kitchen).
You have to be careful about gvinas akum which doesn’t have the same leniency that R’Moshe held of for chalav.
In terms of pots and pans, you would have to be worried about them cooking shrimp, clams and lobster.
RebDoniel,
I think that when it comes to Wine, we are not more lenient for a tinok sh’nishba. The fear is that you will learn from their ways, it is not a punishment. Where is the R’Moshe you are referring to? R’Moshe holds that for someone to be a mechallel shabbos b’farhesia, they must not limit their doing of melacha on shabbos at all (someone who drives, but not in front of the Rav won’t be a mumar).