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Please ask to see the Certificate of Kashrus when visiting all establishments. Current, valid up-to-date certificates are GREEN and they are valid through September 30, 2017.
There are no restaurants that only sell bet Yosef shechita. You would need to check with the Mashgiach at the restaurant that you want to go to to see if they have Bet Yosef shechita meat available on that day. Some butcher shops carry Bet Yosef Shechita at times, but you must check the label.
You would need to check with the Mashgiach at the restaurant that you want to go to to see what kind of meat is available on that day, or what shechita they are using at that time. Restaurants must purchase their meat from KM approved suppliers, and those suppliers carry different brands of meat at different times.
KM certification is reserved for establishments that are Cholov Yisroel, and KDM is for establishments that are not exclusively Cholov Yisroel. KDM certified establishments may have products that are Cholov Yisroel as well.