64 bits may or may not give differences in performance. A 64-bit application can also use (path) more memory than a 32-bit application.
If you run AnyCpu exe on a 64-bit OS, it should run on a 64-bit one (see in the task manager, 32-bit processes are added along with * 32). If you install the application on x64, the library must be either x64 or AnyCpu.
If you do not have native x64 links, you can leave exe and dll as AnyCpu, but you will need to change the setting to x64.
As for the framework, on the x64 machine (which is the only place where the x64 application will work anyway), the framework always includes both 32 and 64 bits found in C: \ Windows \ Microsoft.NET \ Framework and Framework64 respectively.
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