Resolved questions
What is the difference between 격려 and 성원 if both mean encouragement/encourage?

격려하다 / 격려해주다 means “to encourage.” I think the only time 격려하다 / 격려해주다 is used as “to cheer,” is when you cheer athletes/sports team.

i.e.
친구를 격려하는 편지: a letter to encourage a friend.
격려해 주셔서 감사합니다: Thank you for encouraging me.
올림픽 선수단을 격려해 주세요: Please cheer for the Olympic team.

성원하다 means “to support,” or “to cheer loudly.” I have NEVER heard 성원 used for cheering someone loudly when I lived in Korea though.

i.e.
팬들의 성원에 감사드립니다: I thank my fans' support = I thank my fans for their support.