Though there is probably a performance increase with the separation of the Scratch Disks onto a separate HDD, I keep them with with the Project files, so that housekeeping is much easier.
For a separate HDD for the Scratch Disks, I would go with a fast SATA HDD.
The starting configuration is:
C:\ OS, programs and maybe Page File (Windows Virtual Memory)
D:\ Projects (and Scratch Disks in my case)
Beyond that, one could probably get performance increases with a separate very fast HDD for Page File (size can be small), separate HDD for Export and a separate HDD for Scratch Disks. Some separate the Media disks with one physical HDD for Video and one for Audio.
I would anticipate that in a perfect world, one would have a RAID for Video, one for Audio, one for Scratch Disks and one for Export. These would be on a physical controller card with on-board memory cache. You might want to look for the full specs. on "Harm's Beast," as he goes into great detail on his system. Now, he had to place his machine in a commercial walk-in freezer, and keeps the temp around 0 F, to keep it from bursting into flames...
1. Page file: This is an OS / Windows file, nothing to do with PR. If Windows runs out of memory space and wants to continue it's assigned tasks as quickly as possible, it needs to create more space to keep data in memory, but since that is all used, it uses a trick, to make believe there is still room which is virtual memory, Nothing more than temporary storage on a disk, that acts as reserve memory.
2. Scratch / render: When you render a timeline, PR creates disk files of the video on the time line, which include all effects, transitions and overlays. This is in principle done once and shows with a green bar over the timeline. When playing back a timeline, PR uses these created files, instead of the original files and recalculating the effects and transitions again, making for a much snappier playback experience.
Next to these files you have of course the OS & programs, and the original media, and possibly the encoded export files.
If you realize that disk accesses must be spread around as many disks as possibly to avoid congestion or traffic jams, you can understand why the minimum number of suggested disks is three, but possibly more.