Thus, if you cook chicken soup in a saucepan, the pan becomes meat. If you thereafter use the same saucepan to heat up some warm milk, the fleishig status of the pan is transmitted to the milk, and the milchig status of the milk is transmitted to the pan, making both the pan and the milk a forbidden mixture. Kosher status can be transmitted from the food to the utensil or from the utensil to the food only in the presence of heat, thus if you are eating cold food in a non-kosher establishment, the condition of the plates is not an issue. Likewise, you could use the same knife to slice cold cuts and cheese, as long as you clean it in between, but this is not really a recommended procedure, because it increases the likelihood of mistakes. Stove tops and sinks routinely become non-kosher utensils, because they routinely come in contact with both meat and dairy in the presence of heat.
Dishwashers are a kashrut problem.
If you are going to use a dishwasher in a kosher home, you either need to have separate dish racks or you need to run the dishwasher in between meat and dairy loads. You should use separate towels and pot holders for meat and dairy. Consult a rabbi for guidance if this situation occurs. The restrictions on grape products derive from the laws against using products of idolatry. Wine was commonly used in the rituals of all ancient religions, and wine was routinely sanctified for pagan purposes while it was being processed.