Igros Moshe OC 5: 37 says that mikar hadin, there would be no issue with wine touched by a mechallel shabbos. He says that : One may not give wine to a Yisrael Mumar idolater, for he may not drink it. Does the same apply to a Mechalel Shabbos in public? The primary decree was only for real Nochrim due to intermarriage. We find that it applies even to a Yisrael Mumar to idolatry (Chulin 4b). After the decree, Yehoshafat could not drink Achav’s wine. However, if a Yisrael Mechalel Shabbos in public does not serve idolatry, there is no concern for Nisuch. We do not find in the Poskim that he makes Yayin Nesech. The Beis Yosef brought this from the Rashba, and Nekudas ha’Kesef brought it, but they did not put it in the Shulchan Aruch and Shach. It was not so clear to them. The custom is not to drink wine that he touched. A Sefer attributed to Rashi says so. We may decide Halachah based on a custom to be lenient, but not based on a custom to be stringent, for anyone may be stringent if he is unsure. Also, perhaps it is not a universal custom, rather, only for Yir’ei Shamayim. Since the Isur is not clear, and it seems that it is only a custom and not letter of the law, we are not stringent not to give to a Mechalel Shabbos to drink. Even if this is due to animosity, this is the custom. Even if there is a Safek that perhaps some are stringent about this, we are lenient about a Safek mid’Rabanan.
With cheese, if they use vegetarian rennet cheeses, like Tablet K (most cheeses produced in general are vegetarian, as animal rennet is obsolete), one can probably be somech on Rabbenu Tam, Avodah Zarah 35b, s.v. chada; that wouldn’t pose an issue for the kashrus of their pots and pans. Something like mozzarella or feta would be even less of an issue, since it’s a soft cheese that the gezera may not even apply to.
I above said that shellfish and catfish in the pans would be an issue.
You probably wouldn’t have many issues at all eating in such a kitchen.
Also, I try to be makpid on yashan, as that is a real issur d’oraita, whereas the milk we have in America would all be considered halav yisrael according to the Hazon Ish and Reb Moshe.