That is a great question and I wonder the same thing.

"I am going to dump 1/10th of the company that I very publicly acquired if they don't do what I want them to do" feels like manipulation since he's controlling the value of such a large amount of it.

But is it illegal to be willing to lose money if a threat isn't met?

Furthermore, would it be illegal for him to dump his shares, allow the trajectory to tank the price further, then rebuy when it bottoms, rinse and repeat?

It’s not illegal to say “I want to buy more shares at what I believe to be a generous price. If the board won’t sell me these shares, I don’t trust the judgement of the board and will subsequently sell my existing shares.”

That's called a hostile takeover. Part of being a public company.

No, you are allowed to sell your shares if you don't want them anymore.