Generally speaking, will the .NET Windows Forms application run on a 64-bit OS or do I need to change it?
If it does not rely on a 32-bit external library (for example, a COM component), it works fine as a 64-bit process and will use its advantages (large address space, x64 instruction set, ...). If it uses 32-bit things, most of the time you can still run it as a 32-bit application by setting the target platform on x86.
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Most 64-bit operating systems can handle 32-bit applications without problems. That's why you see the Program Files (x86) folder on your 64-bit OS to handle many of your old 32-bit applications.
Until you mix and match library platforms, everything will be fine. Ask x86 when compiling, and you should be good to go.