Is there a specific reason why the answer "Don't do this, grandma!" is not accepted, while "Don't do that, grandma!" is correct?
Not really. The latter sounds more natural to me, whereas the former sounds like she's about to do something really radical. Other than that, not really.
Yes, "don't do that, grandma" sounds like you are (possibly?) scolding her. "Don't do this, Grandma" sounds like she is getting ready to push you off a bridge.
i believe het is mearly understood here, perhaps because this sentence is an imperative, so there's no need to write it. But check with Wicketd (up above here).