"If you have a single threaded application that runs on a server with a multi core processor then does the application only ever use one of the processors?" You don't state how many processors the server has so it could be that there is just one processor but with multiple cores and thus only one processors is getting used as there aren't other processors to use. The warning here is to beware of the difference between the number of cores and the number of processors as there is something to be said for which perspective are you asking the question. For example, a single quad-core processor isn't quite the same as 2 dual-core processors and thus I'd advise caution in your word choice as core and processor aren't interchangeable terms for me.
If there are multiple instances in different processes then they should run in different address spaces in memory would be my guess but that is more up to the OS than anything else.
Something to consider here is what kind of forms are you using here: Winforms or Webforms? If the latter, even though each page may use a single thread, the application pool used by IIS may have multiple threads and thus the code is executed on multiple cores. If it is a Winforms application, I don't know the answer.
Introducing Threading may or may not encourage the OS to use other processors, as there is the concept of Concurrent computing which can complicate the question as there can be the appearance of parallel execution. Thus one could bring in multiple threads but this still gets executed on the same processor possibly.