Is there any concern for the issue of dvar charif for cooked foods? For example, may one use a meaty spoon for a hot dish (pareve) with lots of onions? If this same dish is cold, may one use a meaty spoon to serve it?

According to the Rema (יורה דעה צה:ב) if the majority of a cooked food is a "davar Charif" such as onions for example, if it was cooked in a meaty pot even after 24 hours since its last use the food is considered meaty. If, as you described the onions were cooked in a pareve pot the onions are obviously not meaty and neither do they become meaty by merely serving with a meaty spoon since the onions were not cooked with the spoon. This is as long as the serving spoon was not placed in the pot while on the fire or while still hot. In such a case the meaty spoon would indeed make the onions meaty. Once the onion dish is cold and you are not cutting the onions, you may serve it with a meaty spoon without the onions becoming meaty. Though in general it is always a good idea to serve pareve with pareve utensils to avoid confusion