This question can be directed toward any war where aerial combat was common, but more specifically the Second World War that arguably saw the fiercest and most widespread fighting in the air.
Was there specific insignia placed on an ace's fighter? If so, were they even identifiable during flight?
Were there specific mid-flight maneuvers attributed to certain pilots that made them identifiable?
Would pilots have been briefed prior to a mission that there was an enemy ace(s) operation in their given theatre?
No, not really in the second world war. The speeds and distances at which air combat took place were much much higher than in the first world war. Most skilled pilots did not want to be in a dogfight if they could help it, preferring to bounce and take out unsuspecting enemy aircraft and then extend away, preserving an energy advantage.
There are many anecdotal accounts of pilots engaging aircraft with lots of visible kill markings, and of course pilots could tell when an enemy was particularly skilled or competent, but personally identifying individual enemy pilots was unlikely. Some aces did have personal markings on the aircraft (like the large DB lettering on Douglas Bader's in place of the regular squadron codes), but the flying circuses and distinctive paint schemes and well-publicized aces like Richtofen were mostly in the First World War and not in the second. Pilots would often be aware of specific units stationed near where they would be operating, such as JG26 "Abbeville Boys," and intelligence would likely be aware of the names of leaders and commanders and notable pilots, but pilots would not be given specific briefings about individuals. It's just not how air combat worked.
Fire In the Sky by Eric Bergerud is a great book to look at about air combat in the south pacific, which might give you some insight about this topic.