The Sony a7 III is weather-sealed against dust and moisture, but it is not waterproof. It should not be submerged or used in heavy downpours without a dedicated rain cover.
Yes, it records internal 4K (3840 x 2160) at up to 30fps using full pixel readout with no pixel binning, providing high-quality oversampled footage.
Using the NP-FZ100 battery, it is rated for approximately 710 shots when using the LCD screen or roughly 125 minutes of continuous video recording.
It uses the Sony E-mount. It is compatible with all Sony FE (Full-Frame) lenses and E (APS-C) lenses in crop mode, as well as adapted Canon EF lenses via Sigma MC-11 or Metabones adapters.