Depending on the computer, one can either attach the hard drive to a PATA (IDE) or SATA port and transfer the data themselves. Worst case, you buy a USB enclosure for your old hard drive and install it in there, hook it up and transfer data. I've done that MANY times, once the OS is installed. From what I've read, Windows 7 will run on 1 GB RAM, but prefers 2 GB RAM.
I don't know what happened to your old PC, but if it was BSODing, it could very well be caused by the video card or some other peripheral that has gone bad.