Do CEOs generally hire employment contract lawyers for contract review
and/or negotiation? What about other C-level executives?
Many CEOs have an agent and a lawyer to help them with contract negotiations. Some other C-level executives do the same.
Typically, it depends on the size of the company and the complexity of the contract and benefit package. The larger the company and benefits, the more likely it makes sense to pay others to help review the contract and offer advice.
If yes, at which point on the corporate ladder is it typical for an
employee to hire such a lawyer?
There's no magic point where it makes sense to pay the kind of money required to get counsel. For some VP jobs, it might. For entry-level jobs it probably doesn't.
And there are different levels of contract reviews. It might be very inexpensive to have the family lawyer quickly look over your contract to see if any red flags appear. It would be more expensive to have a contract negotiation specialist review an offer and hold your hand through the negotiation process.
It's a personal decision you would have to make, weighing the costs versus the potential benefits.