He got back home and went to school, so nobody would suspect anything.
Stefan wanted to study so he went to school, he didn't knew anything about Elena.
They met at school, Stefan knew where she studied and got in that school to meet her.
Damon told him to do so.
Season 3 episode 10
Season 1 episode 10
Season 3 episode 19
Season 2 episode 22
Because he got very kind in season 4
Because he needed Elena's blood
Because he loved her
Because he was Stefan's friend and cared about his love