Hi Summer
1. You can use either 저는 세탁소에 옷을 맡겼어요 / 저는 옷을 세탁소에 맡겼어요
Both sentences make sense and meaning the same
2. 에서 comes after "place where the action is happening" In English we may use it by adding at, in, from
Eg. 나는 학교에 간다. I go to school 나는 학교에서 공부한다 I study at school
Hope it helps... You are welcome to ask me any further question
Mia