How could we express the idea that "something that doesn't exist cannot possibly do anything" using a logical argument? And if so, it is at all possible to prove that kind of proposition?
I am not quite sure if I understand what "expressing an idea using an argument" means. I interpreted your question as " please formalise and discuss provability".
Using FOL (First-order logic)
You probably have noticed that in such a case formalising existence with the existential quantifier is no good. This is of course due to the fact that all terms denote something. For example, for the same reason it is inappropriate to formalise "Anton searches Pegasus" naively into FOL: a valid conclusion would be "there is something that Anton searches", but this is of course not true and thus unwanted.
A common solution is to use a unary predicate E meaning "real" existence, while the existential quantifier only signifies the domain of discourse.
E(x) ... x exists. D(x) ... x cannot possibly do something. ∀x(¬E(x) → D(x))
You will notice that I ignored the modalities. I think the intention of "something that doesn't exist cannot possibly do anything" is not so far off from "something that doesn't exist cannot possibly do anything". If you insist that it should be there,
D(x) ... x does something ∀x(¬E(x) → ¬◇D(x))
You will buy all the trouble with quantified modal logic doing that.
Is anything of this provable? Of course not, you need to specify first how the D and E predicate behave.
An important remark
Note what that means: you are ending up imposing conditions on the existence and doing predicate, just to be able to prove what you want to prove. This does not achieve what you want (judging from this and your other questions in this forum). You are putting exactly in what you are getting out.
I think one cannot stress this point enough. There is another quite famous example: you might have heard of Gödel's ontological proof of god. (There is an excellent discussion of this by M. Fitting in his "Types, Tableaus and Gödel's God") The point is basically the same. Gödel introduces a "being positive" predicate in a complicated logic and then proves that there is something that has all positive properties and calls it god. The appropriate response to this is that one hasn't proved the existence of god but shifted the question to "why should I accept this logic?" While such an exercise might be interesting, it doesn't answer the question that one set out to settle.
"something that doesn't exist cannot possibly do anything" is a statement about the world. Logic gives an exact description of what follows from what, which is interesting, but it cannot prove anything about the world. That is simply not what logic was introduced for and one shouldn't use it for that.
I hope this is helpful.
I am not sure that my answer is something you look for (or that is even right) but it may be helpful if I share some of my thought.
Self-evident claims. A claim is self-evident if soon as we understand meaning of the term we immediately admit the claim. For example if I asked you do you know what a triangle is and you answer yes, you admit some set of claims, namely, you admit for example that it must have 3 sides (for if it turned out it did not have 3 side you would say i do not understand then what a trinagle is). So, when you understand something you really admit some set of claims about it. So to answer your question I think that we humans understand what does it mean "to do something (or to act)" and soon as we understand it we accept the claim "there must be really something which is acting". So the claim is really self-evident (or close to self-evident) and that means it can not be proved.
Statment: "something that doesn't exist cannot possibly do anything" seems equvivalent to: "only being in act can actualize a potency". If you were to deny this claim you would deny principle of sufficent reason, because then potency would be actulaized without any reason. And if you accept that thing can happen without any reason, then all of your expirances may happen without reason. And if everything you experinace can happen without reason, then you are not justified in holding to any claim. Therfore you would not be justifed in holding to claim "things happen without reason". So it is imposible to coherently deny the claim you asked about. (I am not sure this argument works)
By way of an example: let's assume that 'being identical to itself' is a property - in fact, the essential property - all things/objects enjoy. Now define a (unary) predicate, say, P, to mean an object is not equal to itself: obviously, no object enjoys this property P. Then you can easily (intuitionistically!) prove
For all x, P(x) implies Q(x)
for any phrase Q you fancy.
So, in fact, it seems more appropriate to maintain that something that doesn't exist can be/do absolutely anything!