Yes it is safe to use these packets for some time after you get them. They are manufactured in bulk and so many times what you get from the store is quite old by the time it gets there. As long as the packets are sealed these can last for months safely.
These can be kept at room temp (mustard, ketchup, salt, sugar, pepper, cocktail sauce, etc.)
I actually save them for later use myself but I normally place the mayo, tarter sauce, etc in the fridge. The ketchup, mustard, and salt/pepper I keep in a bag at room temp for lunches, camping, etc.
They are safe of course as long as they have NOT been opened. Any opened product must be promptly refrigerated.
Also, you must keep the salad dressings cold if you get them from a fast food establishment.
Most packets have a "use by" date on them and are actually safe after this "recommended date".