1. 저는 귀엽단 여자를 좋아해요. '단/다는' in '귀엽단' (= 귀엽다는, contraction of 귀엽다고 하는 = said/reputed to be cute) signifies that the quality of 귀엽다 is not just one's perception but something established more broadly, as by other people. Since '-를 좋아해요' means a strictly personal preference, it should be 귀여운 instead of 귀엽다는. 2. 그 여자는 어려운 시험을 보고 있다. 보다 has other meanings beside "to see". 시험을 보다 is the most common expression for 'to take a test' (there is 시험을 치르다 also, but 보다 is more common). 3. 저의 아버지는 저 먼 집에서 살고 있어요. '저 먼 집에서' might sound slightly unclear if there is no context, but it is not wrong. It is probably saying "in that faraway house", pointing at the house, for example.