Why not "Wat doe je met dat"?
It's simply not used, probably because we have daarmee. :) Similar sentences like Wat doe je met mij? (What are you doing with me?) or Wat doe je met dat voorwerp? (What are you doing with that object?) are used.
when does one use daarmee, ermee, hiermee, or is one of those not even a real word?
They are used whenever you would say in English with that, with it and with this, respectively.
Is it possible to say 'ermee'?
Yes, but only when the object is already the subject.
How is this daarmee "with that" when " daarop" is not "on that"?
Daarop can also be translated as on that, though it may depend on the context and the word order might have to be considered as it does not always work to translate it word for word.