Can software break hardware?

Software cannot "break" hardware. It can significantly increase wear, by, for example, constantly writing to your hard drive, or running your CPU at maximum for long periods of time, which could eventually wear out your fans, but like Michael B said - the short answer is no. All a 32bit OS will do is not let you address more than 4GB of RAM, but that's all. Your hardware is in no danger.

The short answer is NO.

I've had this AMD 64 bit Sempron for 5 years and just installed a 64 bit OS for the first time two weeks ago.

To answer your title question: Software can damage hardware, but it would require some very specialized knowledge about the hardware such that it is likely only to occur when intentionally done under laboratory conditions.

To answer your body question: An x64 processor (x86-64/AMD64/EM64T) is fully backwards compatible with the x86 software. 32-bit Windows 7 will work fine on an x64 processor. You'll just be unable to access memory > 4GB (actually something like 3.3 GB).