Why is 손목시계를한 손으로 차는 것은 어려워요 wrong?
손 한개로->한 손으로 This is better expression.
I agree!
Why is 한 손으로 손목시계를 입는 건 어려워요 wrong? Is there a specific usage for "건" and "게"?
i think this is because 입어 is for wearing clothes and 차 is for wearing belts and watches... although I'm not certain about that, just noticing that pattern
and I also believe that it's "것은" because the "한 손으로" would be the subject, that's why we got a topic marker on the "것".