Yes. While not speaking from personal experience, I have notice a lot of engineers move ahead in management if they have that extra qualification. Be prepared to serve time as a supervisor though, some people are born with better people skills that others.
It had better be something you like doing too. They are entirely two different skill sets.
Depends on what you can afford. The two fields are pretty far apart, even though they are under the same roof (usually). There is a reason engineers are generally considered lacking people skills. Managers, on the other hand, require them. Managers absolutely require strong impulse control, if they are any good.
Combining the two disciplines for a business means the engineer/manager is less likely to be baffled by BS, and can recognize real brilliance when they see it. This makes things more profitable for the company, the core of the issue.