can parve utensils, pots, etc be washed in a meat or dairy dishwasher and still retain their parve status?
YES
Can plastic cups used for cold drinks be considered parve?
YES
Does it matter where they are washed?
NO
what happens to a meat utensil (plastic spatula, fork, pot) that is washed in a dairy dishwasher - and vice versa?
No Problem.
The Rav has a long and extremely thorough Teshuva in his sefer regarding dishwashers.
In a nutshell, there is never an actual transference of good food from one utensil to another. All are transferred via the non-tasty combination of food scraps inundated with soap. Thereby rendering everything inedible. Therefore there is no need to have 2 dishwashers (one for meat and one for dairy).