Hey there, mohrd.
Actually SATA revision 3.0, 6 Gbit/s, 600 MB/s, SATA III are all one and the same thing. If you want to learn more about the different revisions (versions) of SATA, you can check out this link: https://en.wikipedia.org/wiki/Serial_ATA. You should definitely go for a SATA III SSD. I assume that your motherboard has SATA III ports, but even if it doesn't, the SATA interface is backwards compatible and you should have no trouble using a SATA III drive with a SATA II port. Note that this would bottleneck the performance of the drive as SATA II has speeds of up to 300MB/s and SATA III has speeds of up to 600MB/s.
As for the SSD's height, probably a 9.5mm would fit, but that depends on the SSD caddy you're going to get in...