The EOS-1D X is highly weather-resistant and dust-proof with extensive sealing, but it is not waterproof. It can handle rain and splashes but cannot be submerged.
It shoots at 12 fps with full AF/AE tracking and up to 14 fps in Super High Speed Mode (JPEG only, mirror locked up).
No, the original EOS-1D X supports a maximum video resolution of Full HD (1080p). For 4K, you would need the 1D X Mark II or Mark III.
Using the LP-E4N battery, it provides approximately 1,120 shots per charge according to CIPA standards, though real-world usage often exceeds this.