Business School professors - how different are they?
I can address questions 1 and 2 (although for 3, all I know is that my friends say it is very different).
When you are negotiating over salary, your leverage is determined by the best alternative you have. To keep from losing you to another employer, your job must offer a better combination of pay and benefits than anyone else (including benefits like job security, having interesting colleagues and students, having flexible hours and the ability to choose your own projects, etc.).
People in the humanities have very little leverage except from other academic jobs. Mathematicans have more, because there are applications in industry or government, so they get paid somewhat more, but this leverage is limited by the fact that many mathematicians don't want non-academic jobs. Computer scientists have a lot of leverage, and business school faculty have even more.
So basically it comes down to this. Business school faculty often have expertise that is in high demand outside academia and could earn them a lot of money, and there's at least a stereotype that they care more about monetary compensation than some professors do (which makes sense, given their interest in business). This means that to attract excellent faculty, business schools have to pay enough that job applicants would rather work there than in the business world.
In principle, this is no different from other departments: literature faculty also have to be paid enough that they don't leave for the business world. It's just a question of how much that is.
This really depends on the subfield. Business school faculty are certainly more likely to do outside consulting than scientists are, but there are no universal rules (some business school professors do none at all, and some scientists run their own companies). Occasionally, a company will sponsor a university program according to some negotiated agreement, but consulting generally does not involve a pay-sharing deal. Instead, the faculty member simply consults part-time for the company, without using university resources, and is paid directly by the company. In the US, universities typically allow a certain amount of time to be spent on outside consulting, for example one day per week, with no special approval needed.