Your new computer will probably be too different to use the old hard drive as a system drive, but you can use it as a second drive. Get another SATA cable so you can connect one of the empty SATA connector to your hard drive (don't forger to get power from the PSU too). You can either keep the hard drive inside the computer case or outside (careful not to cut the cable with the sharp edge of the case), or plug it to a $15 IDE/SATA to USB adapter, or put it into an empty external hard drive shell case, so you can use it as a backup drive after you have restored whatever you needed from it, provided the old hard drive still works. I would just go with the SATA cable solution, make a copy of everything into an "Old drive" folder or something, and dispose of the old drive or keep it for non-critical usage.

For a new desktop, you can still find Windows 7 Pro computers quite easily, that's the choice I made for a new computer last month. It usually comes with a free Windows 8.1 upgrade disc. I am skipping Windows 8 entirely and will switch to Windows 10 directly when it releases later this year. The Windows 10 upgrade will also be free to Windows 7 and 8 users, although it's not clear for which editions of Windows. Eventually we'll all need to upgrade sooner or later and get used to the new interface.