1) & 3) I agree with tchrist.
1) Your example sentence does not make sense. If it means that 'I' would win the competition because 'I' is more skilled/intelligent than 'you', it would not be phrased in this way. It would more likely be phrased as:
If it were me in tomorrow's competition instead of you, I would win /
If I entered tomorrow's competition instead of you, I would win.
If I were you is almost always used where advice is being offered to modify someone's behaviour. It is not literally what would happen if you were in the other person's place (with your superior skills etc).
eg If I were you I'd take that job / apologise for upsetting him /
mow the lawn today because it's going to rain tomorrow.
@PeterShor's comment If I were you, I would win tomorrow's competition is advice to win the competition rather than lose it. It's not a statement of what would happen if 'I' entered the competition.
2) The only modern instance I can think of where you might use this construction would be in role-playing. Imagine John has an interview for a job, and he's not very good at interviews. Harry wants to help him by pretending to be John and showing him how he says the wrong things. So Harry suggests:
If you be the interviewer, I'll be you.
Another scenario might be hospital staff role-playing so that they get an understanding of each others jobs.
Doctor to nurse: "If I be you and bring the patient in, you be me and
sit at the desk."
Both of these could be said in different ways (Supposing I be you/If I'm you and you're me) but I think you would hear these versions too without them causing any great confusion.