~하기 위해 have the meaning of "in order to", there are some differences in nuance; I guess when
도록 indicates purpose, it can only be used in situations where some type of forcing comes into place. Note that
~도록 하다 has the meaning of "to command". For example:
내일까지는 일을 마치도록 하세요.
Make sure that you finish the task by tomorrow.
In this sense,
긴장을 풀도록 뭐해요? (X)
is awkward since one doesn't command oneself to loosen up. However you could say in this way:
(나는) 학생들이 긴장을 풀도록 농담을 했지만 결과가 시원찮았다.
I told a joke to make the students loosen up, but the result was not as expected.
In this case I am trying to make the students loosen up, so it does make sense. Similarly
지리산을 올라가도록 적어도 2일이 필요하다.
is awkward since there is no special context that someone is commanding another person to climb 지리산.
In the case of another sentences in your link:
길이 미끄러우니까 넘어지지 않도록 조심하렴.
→ The slippery road is making you slip.
네. 선생님께 도움이 될 수 있도록 최선을 다하겠습니다.
→ I am trying to make myself be a help to the teacher.
나는 코미디 영화를 보고 허리가 아프도록 웃었다.
→ This does not indicate the meaning of 'in order to'; rather, it has the meaning of 'until'.
(I watched a comedy movie and laughed until my back hurt.)
On the other hand,
~하기 위해 can be used whenever you want to say "in order to ~". Therefore
긴장을 풀기 위해 뭘 해요?
지리산을 올라가기 위해 적어도 2일이 필요하다.
are acceptable. But
~하기 위해(서) is often used in formal situations, so in the first sentence, to be used with
해요, other forms such as
~려면 makes more sense:
긴장을 풀려면 뭘 해요? (뭘 하는 게 좋을까요?)