Continuity mistake: When Carla turns down the job offer from Kelso the second time there is a pen lying on the table. In the next shot Carla suddenly has her hand on the pen.
Factual error: In the scene where J.D goes to Elliot's apartment and finds Dan there he sings along to "Thank you" by Dido. However that song is from the album "No Angel". The album cover shown in the episode is "Life for rent" which doesn't feature the song.
Answer: No, it was a fake, they used a stamp.
It is real.