Re: Happen to, happen with
About the title, it is important to make your title as relevant to your question as possible; were you asking about the use of 'happen to' vs 'happen with', it would be quite different to your actual question, which is 'do to' vs 'do with'.
As for the two phrases, there is not much difference between them, but I at least see a subtle distinction: "what did you do to your face" is negative, suggesting that something bad has happened to the face; e.g. the make-up looks bad, or there is an injury visible on the face, and while "what did you do with your face" can mean the same thing, intonation would be needed to bring that meaning across – on its own, it would be neutral, more an question than an exclamation. I do have to say that this is merely my understanding of the subtle differences, others, including the teachers, might disagree with me. In fact, I asked a nearby (native English-speaking) friend, and he came up with a completely different interpretation. It rather depends on context and (in speech) intonation.
[Not a teacher]