Does 32bits Windows run slower on a 64bit CPU?

No, and it will run faster than an Windows x64 bit OS.

Mostly all CPUs released in the past few years are 64 bit, but have the ability to run 32 bit code.

During testing when I performed many benchmarks a few years ago on Windows Vista, it was clear that running x86/32-bit operating systems on modern 64 bit CPUs was faster than running native 64 bit operating systems. However this was just starting Windows, when actually in and started, the x64 bit processors showed that they used slightly more memory on applications - however you can also have a lot more, and actual speed for starting up programs was pretty much the same give or take 5% on each platform.

Is a 32bits OS will run slower if installed on a 64bits CPU computer rather than on a 32bits CPU computer?

Depends how fast the CPU is in 32 bit mode. Most modern CPUs are fast enough (as in: no significant speed impact) if I remember correctly. This is why it is the A64 architecture - this is thanks to AMD which realized being ast in 32 bit mode is important.