Yes they do.

The Bing API is actually open to anyone with a bank card but DDG has a specific deal to allow them to filter / sort results, to use ads not only from the Microsoft ads platform, and to skip the redirect link that records what the user click on.

https://www.microsoft.com/en-us/bing/apis

Yes, other search engines can pay for access to Bing's results. DDG isn't just straight Bing results, though, it compiles results from a number of different sources (including their own), with Bing being the primary source.

Yes, they paid Microsoft to be able to access Bing's dataset.