Their meanings are different, but close. So, no, they are not interchangeable but there are situations where any would suffice. #3 is the simplest. It is the same as "What will I do". #1 is an open question. You can use it in the widest variety of situations. For example, you could be walking along a road, come to an unexpected fork in the road, and ask yourself "Which road should I take?" It would not make sense to ask "Which road am I supposed to take?" because you are not "supposed" to take either road in this surprise situation. #2 is best in situations where there is some sort of preexisting expectation or obligation. For example, you could be at a restaurant. You tell your waiter that you want to order the chicken dish that comes along with a choice of two vegetables, but you don't see any vegetables to choose from. You would ask the waiter "How am I supposed to pick the two vegetables" because there clearly must be some way for you to find the vegetables on the menu.

I'd say they're more or less interchangeable. "What should I do?" would probably be a real request for advice. The other two might be as well, but could also be rhetorical questions.