Is it permissible to trust Western restaurants that display "Halal" on their facades but sell alcohol, and is it permissible to eat from them?

The default ruling is that the food of the People of the Book is permissible. One does not deviate from this default ruling except with a definitive proof or a strong presumption. Mere doubt does not render food impermissible. If the ingredients of the food are listed, there is no need for deep investigation and searching, as that would lead to hardship. However, if there is doubt and its permissibility is not strongly presumed, then merely writing "halal (definition)" or "slaughtered according to Islamic rites" is not sufficient. In such cases, a Muslim should limit themselves to fish and seafood or plant-based foods. There is no harm in buying halal food from a shop that sells impermissible items, but it is not permissible to sit at a table where alcohol is served, except out of necessity, while avoiding sitting at it.