Hi. As for the coffee aspect; probably all the keilim (meat and dairy and then some) are all washed together with some kind of soap (that who knows where that comes from) and then on top of everything, I don’t think Tevilat Keilim was on the radar of the establishment’s directives.
Of course, always ask your L.O.R.
In terms of Maras Ayin, I have walked into a non-kosher establishment with my Tzitzis sticking out, reciting LeShem Yichud … invoking the name HaVaYa, Hineini Muchan U’Mezuman LeKayem … the need to use the little boys room.
Most places sell bottled/canned drinks with an appropriate Hechsher.
As with many situations, always ask your LOR and proceed with caution.