All of them are based on 어떡하다 < 어떻게 하다.
For normaly being used in question
어떡해 < 어떻게 해(요)? sometimes used in a difficult situation a talker doesn't know what to do like 나 어떡해! what should I do!
어떡할까 < 어떻게 할까(요)?
어떡하죠 < 어떻게 하죠(하지요)?
어떡하라고 < 어떻게 하라고(요)? ~라고 is used for asking their opinions in rather disapproval like 나보고 어떡하라고! what on earth do you want me to do!
You can use all in the same situation in general for asking a way to deal with sth,
요 is for 높임말 like ~니다,etc.