If a milk spoon was washed together with meat dishes, is it still Kosher? If a Parve pot was put in a dishwasher with milky cutlery, does it become milky?

Washing parve dishes
Keeping things straight in a kosher kitchen can get tricky sometimes, especially when one want to keep something parve, and not just to separate milk from meat. In general one should make an effort to wash up the parve dishes in a sink by themselves, with a separate sponge. This way, even though they are being washed up in a sink that was used previously for milk or meat, because the sink was emptied and washed out, there is no chance of any of the milk or meat being transferred to the parve dishes.
If parve dishes were put together in the sink with (say) meat dishes, it could be that the meat taste transfers to the parve dishes. Though this would be a rare occurrence - needing a hot meat pot to spill over to the parve dish, or a hot piece of meat to fall onto the parve dish etc - it could happen. In such a case, I would advise to call a Rabbi and tell him the exact details.

The wrong sink
If a milk dish was washed in the same sink with meat dishes: if all contact between the dishes was made when the dishes were cold, everything is fine, and nothing needs to be koshered. One should just now clean the dishes being careful not to use hot water.
If the dishes where washed with hot water: if the hot water was poured onto both a meat and the milk pot, one should consult a Rabbi as the law depends on many factors (such as if the dishes were clean or dirty, had been used in the last 24 hours or not, if there was detergent on the dishes etc).

A dishwasher
What is the Halacha when (for example) a meat pot was washed in a milk dishwasher?
In order to answer it fully it would be helpful to have a bit more information, such as - was the meat pot dirty or clean? Had it been used for hot meat or cold food? Were there dirty milk dishes in the machine at the same time? Did you use washing detergent? Here is the Halacha for the common cases:
There is some debate amongst the Poskim as to how to view a dishwasher – either as a first vessel, or as water poured from a first vessel, or as a second vessel. The way this question is resolved will affect the ruling of mixed dishes in a dishwasher. There is also an argument about the strength of detergents today. And lastly, there is an argument between the Shulchan Aruch and the Rema about the law concerning milk and meat dishes washed together in a first vessel (see Yoreh Deah 95,3).
Therefore, the strictest ruling in such a case would be to consider the dishwasher a first vessel, and rule like the Rema, and (if all the vessels were ben-yomo, i.e. used within the last 24 hours) rule that everything needs koshering (or at least the meat pot - depending on if there are 60 times its size in the other pots and dishwasher, which is very likely). However, there are opinions that are more lenient - especially when the dishes were rinsed, and the detergent is strong (see Yalkut Yosef, Otzar Dinim L'Isha U'Bat, p. 303-308).

So if the question is if there is a concern - the answer is yes, there are certainly opinions who are concerned about this issue. However, there are certainly opinions to rely upon, after the fact, and consider everything as kosher. Even though one should not put meat and milk utensils in the machine at the same time, if one did so, after the fact, one can rely on the lenient opinions and rule that everything is still kosher.

This is based on the assumption that the dishwashing detergent is strong enough to annul any tastes that come out of the vessels, or small remains of food in the machine. Together with other halachic considerations, many rabbis rely on this ruling, especially when not all the facts are available.