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
I can recommend VirtualBox for this purpose, you can download the free version and it's easier to use than VMWare. However you will need a 64bit installation CDROM, and storage space for a full system install, and if you are installing Microsoft Windows they will expect you to pay for a license key.
Also your CPU needs to support both 64 bit mode in the hardware, and the vt-X/AMD-V features (most of them do). It's a bit slower, although mostly that seems to be the display that slows it down, not the internal program calculations. This is NOT an emulation, the CPU is really running native 64 bit, but VirtualBox fakes the hardware devices (display, disk, network, etc) so the result is not as nice as running normally. 3D graphics acceleration is available, but it has limitations.
An easier option is simply to hire an online Virtual Machine by searching for someone offering 64 bit versions of Windows Server (there are plenty). Usually they will offer connection over Remote Desktop, typically you can pay by the month. Upload your programs, run what you want, then delete it when you are finished and cancel the service. The service provider handles installation, licensing, hardware, etc.