None of the stuff from Special Stages is recorded. Nor can you unlock them in any other way but through a normal level.
Also you get shown your time at the end of Bonus Challenges, but you have to be damn quick to write it down. Both Bonus and Emerald Challenges total your score at the end, leaving it up for a short while, but not indefinately. Also on both you get an extra life for every 10000 points.