- 의문 = a question formed in one's mind. A question explicitly asked is called 질문. - 의심 = doubt or suspicion in one's mind. 의심 is thus stronger and more negative than 의문. When expressed out loud, it's likely to becomes a criticism(비판). - 의심의 여지 = room for doubt (This phrase by itself has no sense of "no"). - 의심의 여지가 없다 = There is no room for doubt. X-가 없다 = "there is no X". - 의문의 여지가 가튼 듯이는것같아요 => '의문의 여지'와 같은 뜻인 것 같아요. (quote marks are optional) 의심의 여지 is more common than 의문의 여지. Since a mere question (의문) is a pretty innocuous thing, saying there is no room for question might be a bit too strong.

I think the word "의심의 여지" is better than "의문의 여지" for me I don't mean "의문의 여지" is wrong. I can find sentences both "의심의 여지" and "의문의 여지" and "의심의 여지" are used a lot more than "의문의 여지" If you wanna ask me what the difference is, there is no difference in each meaning. The reason why I say like this is that the former sounds just better for me. And as i think that's because I heard the former more than the latter. PS I don't understand what you want in your PS. I can just revise it to be correct one. 그 문제 그렇게 제가 했으면 됐어요. 이 문제도 이렇게 못 할지도 몰라요. But this doesn't make sense. If I translate it, it is I was able to handle that problem in this way. This problem might not be able to be handled also in the very same way. It sounds weird, doesn't it? 그 문제 그렇게 제가 했으면 됐어요. 이 문제"는" "그"렇게 못 할지도 몰라요. I was able to handle that problem in this way. But, this problem might not be able to be handled in the very same way.