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Do the companies provide parts for free (or with heavy discounts) in exchange for having their logo on the car?
Or do they pay to have their logo on the car, and if so, can a car use Pirelli tyres but have a Bridgestone logo on the car, because they paid more?
Finally, do the teams approach the companies (by going after who they think makes the best parts for their car), or vice-versa (the company going after the team/car they think will give them a better return on advertising)?
It varies by the situation. Some teams send requests to companies to sponsor them, while other times companies ask teams to represent them. Usually it is the latter. In F-1 companies such as MacLaren, the company has ties with tire and parts companies and therefore is able to use those logos and get their products for a greatly reduced price. Some cigarette companies or condom companies would also put their logos on Formula 1 cars in the early years, but not anymore.
The question has a lot of varying answers. A car can have sponsors, the event itself has sponsors and a driver can have sponsors. As noted previous, the driver's sponsor will have badges on their racing suit (and likely require a mention if the driver is ever interviewed).
Car and event sponsors will have places on the car. Generally, car/team sponsors will not allow competing manufactures to "share" space on their car (which is usually specified in the agreement). So in your example, a car with Bridgestone badge on it, will be running Bridgestone tyres. And not allowed to have another tyre manufacturer badge on the car. Unless...
The exception will be if a competing manufacturer sponsors the event itself. In your example, if Pirelli sponsored the event itself, they will require a badge be on every car participating in the event. You can usually identify an event sponsor over a car sponsor by the placement of the badge. If it is in the same place on all the cars in the event, they are likely an event sponsor. If Bridgestone is only on a few cars, they likely only sponsor those cars.
You can also generally determine how much support a company has contributed to the event, car, driver by how large the badge is in comparison to the others. More space, larger badges equals more support. Keep in mind that this will be a ratio. The actual amounts will vary by event, car and driver.
The support itself could be outright cash, or parts (free or reduced prices) or anything the sponsor and team agree upon. It could also be something the team/driver support in some way themselves (i.e. a non-profit of special interest).
