Add air conditioner climate entity to Home Connect

[Diegorro98]

Proposed change

Add air conditioner climate entity to Home Connect

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 climate entity for air conditioner to Home Connect core#155981
• Link to parent pull request in the Brands repository:
• 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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[Copilot]

Pull Request Overview

This PR adds climate platform support to the Home Connect integration, enabling control of air conditioner appliances through Home Assistant's climate entity interface.

• Adds climate platform to the integration's supported platforms
• Documents the new climate entity for air conditioner control
• Specifies supported controls including HVAC modes, preset modes, and fan speed

[frenck]

✅ Approved. Can be merged as soon as the parent PR gets merged.