Thanks for the challenge. I always have fun thinking about things like this. You asked,
"Can you prove that "nothing happens" is necessarily false ?"
No. I don't think I can prove anything except in the context of formal mathematics. Even then, I am not all that competent. In order for me to prove anything in mathematics, I insist that all the primitives be spelled out explicitly, that the axioms be listed, that each term other than the primitives be defined, that all the notation be explicitly defined, and that a sufficient set of proved theorems be known to me. Only then might I claim that I could produce a proof of some proposition.
Now, in a weaker sense, you might be asking me if I can produce an argument that would convince you that "nothing happens" is necessarily false. My guess would be that I couldn't do that either, since you strike me as a guy who might be hard to convince.
Weaker yet, maybe I could convince at least someone that "nothing happens" is necessarily false. Or maybe I could just convince myself.
With one of those more limited objectives, I guess I would argue thus:
To show that "nothing happens" is false, it is sufficient to produce a single counterexample. A counterexample would be something that happens. I know from direct personal experience that thought happens. QED
Now, you tacked on a baffling parenthetical assertion:
"(if "thought" is mutually exclusive with "nothing" then the two sentences can't possibly be true at the same time)"
Your premise states that "'thought' is mutually exclusive with 'nothing'". That is hard for me to swallow. Thought and nothing are concepts of a different type. The relationship of mutual exclusivity is meaningless between them. It would be like declaring that apples and justice are mutually exclusive. I can have apples and justice present or absent in any of the four combinations in different particular situations. Similarly, I can have thought and nothing present or absent in all four combinations: 1) The astronauts may have left a box with nothing in it on the moon. In that vicinity, there is no thought and nothing in the box. 2) While they placed the box there, the astronauts were thinking, so there was thought and still nothing in the box. 3)...4) you get the idea.
So, since I deny your premise, I can't logically arrive at your conclusion. It was fun trying, though. A counterexample to your conclusion would be that thought happens and there is nothing in the astronaut's box, so nothing is happening inside the astronaut's box.
Thanks for the fun, and