Add documentation for CatGenie integration

[kclif9]

Proposed change

Contributing doucmentation for the new CatGenie integration in Home Assistant

Type of change

• Spelling, grammar or other readability improvements (current branch).
• Adjusted missing or incorrect information in the current documentation (current branch).
• Added documentation for a new integration I'm adding to Home Assistant (next branch).
  • I've opened up a PR to add logos and icons in Brands repository.
• Added documentation for a new feature I'm adding to Home Assistant (next branch).
• Removed stale or deprecated documentation.

Additional information

• Link to parent pull request in the codebase: Add CatGenie integration core#169533
• Link to parent pull request in the Brands repository: Add CatGenie logos brands#10226
• This PR fixes or closes issue: fixes #

Checklist

• This PR uses the correct branch, based on one of the following:
  • I made a change to the existing documentation and used the current branch.
  • I made a change that is related to an upcoming version of Home Assistant and used the next branch.
• The documentation follows the Home Assistant documentation standards.

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Pull request overview

Adds a new integration documentation page for the CatGenie integration, describing supported devices, setup prerequisites, provided sensor entities, and a basic automation idea.

Changes:

• Added source/_integrations/catgenie.markdown integration page with front matter and introductory content.
• Documented supported/unsupported CatGenie models and prerequisites for setup.
• Listed provided sensor entities and included an example automation scenario.