There is usually yes. I think complete separation would be the exception – unless enforced by the local government, to prevent monopolies or insider advantages – to give an example would be Microsoft, they’re supposed to keep their operating system and office departments separate! (I can’t see how that could be policed effectively though!)
Yes definitely, especially in a manufacturing company like an F1 team.
The work flows through a company and so alot of departments work with others. For example before a car is raced, the mechanics build the car, but the parts need to be manufactured and before that they need to be designed. This isn’t always a straightforward flow of work either, there can be various stages where it goes between different departments.
And before any of that the marketing and finance departments need to have the money available to pay for everything!