Honestly, I think the difficulty you're having is one of form, not content. That is, you understand the mechanics behind the proof, but you don't accept that the form of the proof works. Your objections hinge on the implications of a paradox, and our seemingly arbitrary construction of Q.
Instead of banging your head against this proof more, I think you may want to read up on proofs in general. Your objections would apply to any proof by contradiction.
Example: consider that Q exists even if we don't define it. We discovered it, not invented it. Given P, we can define Q', but the problem with Q is still there. But this really has nothing to do with this specific proof - it's more fundamental than that.
Instead of banging your head against this proof more, I think you may want to read up on proofs in general. Your objections would apply to any proof by contradiction.
Example: consider that Q exists even if we don't define it. We discovered it, not invented it. Given P, we can define Q', but the problem with Q is still there. But this really has nothing to do with this specific proof - it's more fundamental than that.