No, this is an EF-S lens designed exclusively for Canon APS-C sensor cameras. It will not mount on full-frame Canon DSLRs.
Yes, the built-in Image Stabilizer helps smooth out handheld video footage, though the micro-motor autofocus may be audible in quiet environments.
The IS II uses a DC motor for focus, while the STM version uses a Stepping Motor for near-silent, smoother autofocus during video recording.