Inevitably products that are on sale pop up all the time like cars, whisky and phones. If the props department happens to use an Audi, Jack Daniels or an iphone, do they get in touch with them and see if they can get them to pay for it? Or threaten to use their competitors products?
Free promotion for undisclosed reasons
You'll see plenty of movies where characters use Apple products, however Apple have claimed (via the Washington Post) that they aren't paying for it:
Though DailyTech claims they achieve this by giving the products to those producing the film:
Parodies and references
We've also had a related question about mocking companies in Idiocracy, and from the interview posted it sounds like they weren't paid for the (almost) use of their names.
There are also cases where a studio can advertise their own products for free, for example Sony Pictures Entertainment (and their child company, Columbia Pictures) often show off Sony products in their own films, such as The Smurfs and Casino Royale.
Other examples would be when a character is watching TV (usually the well-timed news broadcast) and the News channel is one that the studio own. For example Fox's 24 will only ever show Fox News or Sky News (in Live Another Day), both of which they own.
Presumably, yes, if they need the money. Financing films is hard, and independent film-makers have nothing to lose by trying, though I have found plenty of articles saying it's very unlikely companies will be willing to contribute to a film unless they have a reasonable expectation of making their money back (for example having famous people working on the project that will draw an audience).
I worked for Ben Affleck's studio LivePlanet (Project Greenlight, The Running Man) for a year. We did have a few sponsorships for each of our shows. In general, we just went to great lengths to exclude any brand references, because in made-for-television content, conflicts with sponsorships can be a problem.
For example, if one of our shows happened to pan across a Coke can, but the television network had a lucrative Pepsi ad deal, we might have to go back and blur the can or even edit out the scene. It's much easier to just keep all brand references out.