It's reasonably safe, though we can offer no guarantees. Plenty of people have flashed drives like that for years, but modifying a console is always AYOR.
Yes, just use the SATA cable. Plug in the drive, power on the 360, then power on the PC. If it shows up in Jungle Flasher, you should be good to go. Getting it recognized might take some variation in method (power up 360 after PC, starting JF before or after drive power-on, or any combination of methods), so you're going to have to just try it and see what works.
VIA is a chip manufacturer. Traditionally, SATA cards with VIA 6421 chipsets are very good with allowing drives to be recognized and flashed, but, like I said, sometimes it's hit-and-miss and the drive just doesn't want to be recognized.