No. This is an EF-S lens designed specifically for APS-C sensor cameras. It will not mount on full-frame Canon DSLRs like the 5D, 6D, or 1D series.
Yes, the Image Stabilization helps significantly with handheld video, though the autofocus motor may produce audible noise that an external microphone can help mitigate.
The IS II uses a DC motor for focus, while the STM version uses a Stepping Motor for smoother, quieter autofocus during video recording. Both offer similar optical reach.