I'll add some of my own ideas: The particle "을" after 것 is wrong. It should be 이. 것이 can be shortened to 게. 을/를 is used with 좋아하다 but 이/가 is used with 좋다. Further, as Oya sorta pointed out, there are set expressions for "What should I do". Hmm...I can't think right now...어떻게 해야 하죠? for example. Or 할까요? Or 하나요? Those are all right.

"뭐를 하는것을 좋을까요?" => 뭐를 하는 것이 좋을까요? (을 -> 이) Your sentences are good except for the particle 을. Only "이" is acceptable after 것 here because 뭐를 하는 것 is the subject. 을 is out of the question, and even 은 is not appropriate here because it is asking about something specific. Related examples: - 이것을 하는 것이 좋겠다: It'd be good to do this ("doing this" would be good - "doing this" is the subject). - 무엇을 하는 것이 좋을까?: What would be good to do? (What should I do?) - 시골로 가는 것이 좋겠다: It'd be good to go to the countryside. - 어디로[에] 가는 것이 좋을까?: Where would be a good place to go? (Where should I go?) * Comparison of "무엇을/뭐를/뭘"(what) and "어떻게"(how): - 뭐를 하죠?: What should we do? (focus is on 무엇 - we don't know "what" to do out of many possibilities). - 어떻게 하죠? == 어떡하죠?: What (are we) to do? == How (by doing what) should we respond to the situation? (In English "what" is correct, but in Korean it should be "how"(어떻게), similar to "How can we get out of this?") This "what" vs "how" comes up often: - What do you think of him? = 그에 대해 어떻게 생각하세요 (무엇을 is never used in this context). - What happened? = 어떻게 된 거야? (뭘, 무슨 일 will be unnatural here).

when korean people want to say "what should i do?" (according to what I have heard ), they say "어떡하죠?"