Proving the existence of nothing renders it as something and therefore it is no longer nothing and doesn't exist. Consequently, if you can prove that nothing doesn't exist then you have brought it into being.
"If Something is Nothing, then Nothing is Something..." (around and around we go)
As to the phrase "Do Nothing", there needs be an implied qualification, scope, and/or definition that is understood by the speaker and listener. Nothing must have an understood definition of the
thing that you are doing none of (as CajunCenturion has pointed out above). In its rightful context, it is with some certainty that I say an entity can do nothing.
It should be noted that the qualification, scope, and/or definition in this case is coming from the questioner, and not the one who is answering. Consider the following...
Question: "What are you doing?"
Implied: Tell me what you are in the midst of doing that would be of any interest to me?
Answer: "Nothing."
Implied: "I am not in the midst of doing anything that I believe would be of any interest to you."
If you take away the implications, as has been the case in more than a few posts above, then the answer does indeed become impossible and a falsehood. However, to remove this additional context that surely frames the question changes the intent completely and you are no longer talking about the same thing any longer. You have abscribed a very narrow and unintended context to the original question and forced the answer into the explicit in an attempt to make your argument.
Question: "What are you doing?"
Implied: "Tell me everything, down to the minutest detail, that you are doing at this very moment?"
Answer: "I am breathing, blinking both eyes, living on earth, thinking about your question, answering your question, existing in the moment, talking to you, looking at you, listening to everything the my ears can hear, ..." (around and around we go)
While it is true that an answer of "Nothing" to the above question would lack veracity (if the implied was understood by both parties), it is also true that an argument that only holds up in such circumstances could be discounted in the vast majority of situations in which one finds themselves presented with such a question.