Why was Jade awake before entering the Medium? Was it her title, or was it some aspect of her personality like Dave/Dirk? Also, were any trolls/post-scratch kids awake before entering the Medium?

So far every Space player has awoken early, so it was most likely her aspect that allowed her to be awake.  The character known to have been awake before their game started are Jade, Dave, Dirk, Calliope, Terezi and Kanaya