If I put pareve containers that have only had cold food in them and wash them in the milky dishwasher does it make the container milky?
No, they retain their pareve status.
It is questionable if they received any dairy flavor from the dishwasher. Even if they did it would likely be spoiled by the soap. Although we are stringent with regards to milk and meat, with pareve containers, the flavor would only be secondary [nat bar nat] and so there is more room for leniency.