꼭 indeed has those meanings you listed.
exactly, with no difference:
네 얼굴이 꼭 화난 것 같다: Your face looks like you are (just like) angry.
네 말이 꼭 틀린 건 아냐: It doesn't mean that what you said is wrong, per se.
surely, definitely:
너 선생님께 꼭 전화해라: Be sure to (don't forget to) call your teacher.
이 문제는 꼭 일어난다: This problem will arise without fail.
tightly:
문을 꼭 닫아라: Close the door fully (tightly).
병뚜껑껑을 너무 꼭 닫아서 안 열려: The jar's lid has been closed so tight I can open it.
꼭 꼭 숨어라: Hide yourself deep and tight. (children's hide-and-seek play word)