> Is the CEO out in the market calibrating salaries every day?
HR should not be setting compensation. That's a business question and something the business people need to understand and deal with.
> Does the CEO have time to analyze optimal organization designs?
Organizational structure is also a business question. It goes directly to what sorts of team structures, etc, are most effective in delivering the company's product.
> Should the CEO do all the initial fit screens?
Not necessarily the CEO, but some technical person should do the initial fit screens.
> Chase down background checks?
This is the sort of thing HR should do.