Removing HD from PC and putting in another PC
You can always take an existing hard disk and put it into another machine. To do that, just unplug the power and data connection from the disk, and plug it into the other machine.
If both machines are the same age, chances are good you have the same connection type for the hard drive. Judging from your post it seems like that's okay. If they aren't, you won't be able to just plug in a new drive into an old machine and vice versa, that's because hard disks now mostly use SATA which isn't available on old motherboards that still use IDE/PATA.
But: You won't just be able to run the existing operating system if the hardware specifications are not the same. That's because your installation has specific drivers for your hardware which don't work out of the box for other hardware.
Theoretically yes, but in practice it's a little more complicated.
The drivers installed on the hard drive will be for the original hardware - motherboard, disk controllers, CPU etc.
Windows will attempt to sort things out but if the hardware it too different (e.g. the motherboards are different) it's most likely that it will fail to boot.
If you just want the data off hard drive install it as a second drive in the new PC. Remember to switch the Master/Slave jumpers on the disk first.
It will most likely not work. This is because the Cpu and some other hardware. It is possible that it will boot but you will see errors and it will never run smoothly.
There is however no harm in trying this. You can always put the disk back in the original computer.
The physical procedure is simple if the hard disk is compatible(ex. the HDD has an IDE and both machines have IDE ports). However in order to boot windows on one that was installed on another PC, you will need to create a separate hardware profile. Here is a link to helpful article.