[Avodah] Kashrus Question

[Prof. Levine]

As is well known, there are rabbis and rabbinical organizations that 
give supervision to stores owned by non-Jews that are open on 
Shabbos. Indeed, some of these stores are open 365 days a year. There 
are a number of Dunkin Donuts stores located throughout the country 
that are like this.

What is the kashrus status of coffee that the gentile store owner 
brews on Shabbos that a non-observant Jew who comes into the store on 
Shabbos afternoon drinks? What does it mean to say that Rabbi so and 
so or organization such and such supervises this store regarding 
other products that are heated up on Shabbos?

What is the kashrus status of donuts sold during Pesach at such a store?

One can ask similar questions about the plants that produce Thomas's 
and Arnold's baked goods during Pesach. I believe, but I may be 
wrong, that these products, having been baked on Pesach, appear in 
stores during Pesach with an OU on the label. If so, what does the OU 
on the label imply?

I am simply asking for clarification and wait to be enlightened.

Gut Voch.

Yitzchok Levine