yes it can, but this doesn't mean 'to position oneself' (most common usage). 'to position oneself' is to take a stance, either literal or metaphorical; e.g., on a playing field or against a political policy. it simply means 'to put/place oneself' on a diet.
You will not be put on a diet. I thought mettere means to put and reflexive version to put yourself?
You would more likely say, "Don't put yourself on a diet". The translations are not common vernacular.
Yes, but you still have to use future tense in your translation of their sentence. So, "You will not put yourself on a diet!" or "You are not going to put yourself on a diet". Your sentence "Don't put yourself on a diet" would be "Non metterti a dieta!" in Italian.