Yes, it records 4K DCI and UHD internally to CFast 2.0 cards using the XF-AVC codec at 10-bit 4:2:2.
Using the standard BP-A30 battery, you can expect approximately 125 minutes of continuous recording. The larger BP-A60 doubles this to roughly 250 minutes.
No, the camera is weather-sealed against dust and moisture but is not waterproof. It requires a dedicated rain cover or underwater housing for wet environments.
Yes, the EF mount on the C300 Mark II is compatible with both full-frame EF lenses and APS-C/EF-S lenses.