The reason, I think, is that these are forms of embedded questions. If you said "I don't know where she lives. Do you know?", that would make sense. But if you said "She lives in Tokyo. Do you know that?", my response would be "Yes, you've just told me!", hence using "Did" for questions where the information is given.
It's a bit like reported speech, where even though something may be true now, it must have been true before you spoke, hence the past tense, e.g. "I said she lived in Tokyo".
You could even say "Yes, I knew that, " in response to "Did you know she lived in Tokyo now?" but it would be incredibly normal, although grammatically lazy, to say "Yeah, I know"