If the processor in the computer is 64 bit - meaning it was designed to run 64 bit instructions - then it is ABLE to run 64 bit OS. But it doesn't have to. Running the 32 bit version on a 64 bit system will not cause your kind of problem. The 64 bit version would not install on a 32 bit system, I believe.
But you do need to know which version of Windows 7 you installed in order to get and install the correct drivers for your system. Drivers will come in 32 or 64 bit flavors.
Yes, the chipset on you motherboard performs all of the necessary functions to operate the CPU, RAM and hard disk controllers. The chipset drivers contain all of the instructions needed to run those chips at the manufacturer's specs. Sometimes the generic Windows drivers are good enough for basic use, but many times the manufacturer's driver is much more complex.
You need to know the make and model of your motherboard (and all the other connected components that require drivers) in order to go to the manufacturer's website and download the correct drivers in the correct bit rate.
You will need to do this separately for each of the systems you are upgrading.