Just a simple question: Is there any way to run a program compiled under a 64 bit Windows environment (with mingw64) on a 32 bit machine? Any DLL or any compatibility layer
No you cannot do this. The other direction is possible through an emulator, e.g. on Windows it is called WOW64.
It is standard practise on platforms that still have large install base of the 32-bit versions of the OS to ship either just a 32-bit version, or both 32- and 64-bit versions.