If there's some code I'd must write natively to bind the Windows app to the Linux system (e.g. interacting with some low-level Linux stuff), can I build that library as a DLL with wineg++ / winelib and have the WINE-invoked app use that?
Yes. This is what I meant by a bridge between Linux and Windows code.
The easiest way to do this is with LoadLibrary and GetProcAddress, but linking to a winelib dll is also possible.