Can I use the same lid for both dairy and meaty pots or pans? I wash my parve utensils together with the meaty dishes; if I dry them afterwards with the meaty towel (and not with the parve towel), do they get meaty?

Using the same lid for both dairy and meat pots as the food is cooking will in many circumstances make both the pots (together with the lids) and the food not kosher. Even using the same lids to cover meat and dairy food stored cold in the refrigerator is against Jewish custom that requires separate utensils for meat and for milk (Aruch HaShulchan Yoreh Deah 89 16). If you wash the parve utensils always using soap and making sure that they're not directly in contact with meat, they remain parve.