Got it and Get it both is used to mean do you understand? The person who asks, do you understand? can use both of them but more often they use get it. And the person who understood could use only 'got it' to answer.
If you mean one cannot answer "get it?" with "get it", I'm afraid you're wrong. I've seen and heard it used by native speakers. "(Do you) get it?" - "(Yes, I) get it." Pretty natural, I think.
You also mean, I guess, that "Got it?" may imply "did you write everything down?" Then, can it mean "have you done everything?" as well? A more general meaning?
I would think I would reply to "did you get it?" with "yes, I got it it" whether referring to an object or a question. If asked "have you got it?" I would be asked about having a particular object, such as a book or a job, or perhaps understanding an explanation.