1. when you speak to yourself you use it '지' at the end of verb. well.. I cant explain exactly, "어떻게 신청하면 되는 거지?" if you spoke it to someone in the same meaning, you would speak like this "어떻게 신청하면 되는 거야?/어떻게 신청하면 돼?" in the same way, "왜 아무 말도 안 하지? (to yourself)" can be if to others " 왜 아무 말도 안 해?" 2. '~(하)다가는' is used when, especially, assumption in wrong situation. it means 'if you do(keep doing) that, it might get in trouble. It entails bad situation. We usually use '잘못하다가는(if the situation get worse)', '자칫하다가는'. ex) 잘못하다가는 다칠거야. if you do something wrong accidentally, you may get hurt.

1.in this case '지' is not single word. it's use with ~안 하지, ~되는 거지, 나가지, 줘야지.it's like suffix 2. ~다가는 means if some action dosen't stop, the action will change to another action in your sentence 잘못하다가는 부모님 뵙는 시간보다 차 안에 있는 시간이 더 많겠어. If we wait too much, like this(if dosen't stop action) we would have stay more time in the car than meet father and mother(would be changed another action). 3. I think 수공요 it's miss typo. he or she want to typo 수고요. it's means 수고했습니다.