Are there any Windows7 parameters that control the creation of a .dmp file?
There are parameters which control the creation of a crash dump: see MSDN on Collecting user-mode dumps.
Are there any requirements/flags that need to be compiled into a program in order for it (or windows) to create a core dump file if it crashes?
You don't need to compile anything in for the previous answer to work. However, the program needs to terminate due to an unhandled exception, which means you need to let the exception bubble up and not being handled by the unhandled exception handler.
I'm 99% sure it must be Windows that is responsible for creating the core file, since the program itself would be dead/terminated when it crashed, correct?
As stated above, Windows can handle that and it's a good idea to have Windows handle the crash. Imagine that your program is broken due to a memory leak. Some memory has been overwritten. In that case, your unhandled exception handler can be destroyed. Windows, however, still has full control over the process, can suspend it and create a dump from outside (rather from inside).
Are there any other things I should be aware of, or check for, or 'evidence' I can collect and then show our developers?
Well, suggest letting the dump be created by Windows due to above reasons. Then they also don't need to implement a configuration setting (you don't want the crash dump file to be always created, do you?). You don't need to implement a limiting number for the files. You don't need to implement a check for disk space, etc.
And you can suggest to read the Windows Internals 6 books.