I should know betterI should probably keep my mouth shut, because this whole thing is extremely definitional and, consequently, colloquial to the point of idiosyncracy.
As to the first question, is smoked salmon cooked? It depends what you mean by "cooked." As RPM pointed out, It's cured in a salt-acid-sugar brine, then cold-smoked. It's not a raw product in the sense that sushi is raw, and it's not cooked in the sense that it's internal temperature has been raised to above the point required to kill most native bacteria and parasites. However, the brine and smoke do alter the cellular structure of the fish, and it will keep much longer than raw. No matter what side of the argument you take, you're right.
Suzanne's description is all over the map, as I understand the terms "indirect heat" and "barbecue." The term barbecue is so elastic it means pretty much whatever anyone wants it to mean -- and consequently becomes is meaningless. Even among the strictest of the strict there is some argument as to whether "barbecue" includes direct-heat grilling -- which is the most common usage. The majority of purists insist that "barbecue" is only indirect heat plus some amount of smoke. I do a lot of hot smoking myself, and can assure you that the heat is, indeed, indirect and the product barbecue. Also, when you go to a "barbecue" restaurant you expect smoked meat. OTOH, "open pit" and "California" (or "Santa Maria") barbecue include some degree of direct heat.
Scientifically (and precisely) we would say that heat conducted by air convection is indirect heat; but heat transferred by immersion, contact conduction, or radiation is direct. Does this help? No.
RPM's search result has the right of it as a matter of process, but while insisting the lox is not cooked, fails to define the term.
Abe has me looking very much askance. "Lox" is not only eaten by Jews, in fact, I'd venture to say that Jews are a small but noisy minority (what else is new?) of cold smoked salmon eaters -- compared to Europeans, especially Eastern, Scandinavian and British Europeans. Add gravlaks to the mix, and the proportions become more imbalanced. Lox is not only taken from the belly, or even particularly -- although belly lox is highly prized for its high fat content. Kosher is as kosher does. It's hard to handle a scaly fish like a salmon that makes it treif (not kosher), that includes nothing involved in lox making. No bacon here. Furthermore, if anyone chooses "kosher" for health reasons, they aren't getting much if any benefit compared to any other system in which meat and poultry are properly handled. In fact, "organic," "free range," "CAB," "grass fed," etc., are all usually better indicators of good practice.
The owner's religious and ethnic identity have nothing to do with the disclaimer. It is also unlikely that fear of liability is at issue either. Speaking as an attorney with litigation experience in the area, unless the sign is mandatory it would act as a red flag in most law suits -- in that the seller knew the product was dangerous and sold it anyway. Most likely, the state or municipality required the display.
Then there's one more term:
Abe, by "Hebrew," do you mean "Jew" and "Jewish?" If so, I've got to say that "Hebrew" is not commonly used in those ways; and is mostly associated with anti-Semites. As a Jew myself, seeing it here makes me uncomfortable. I'm sure you're not anti-Semitic, don't mean any harm, and now that you know will stop.