Add initial Hot Spring integration documentation

[Moustachauve]

Proposed change

This is the initial PR that introduces hotspring integration documentation.

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 new Hotspring Integration core#177992
• Link to parent pull request in the Brands repository: Add Hotspring Brand brands#10898
• 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.

[home-assistant]

It seems that this PR is targeted against an incorrect branch. Documentation updates which apply to our current stable release should target the current branch. Please change the target branch of this PR to current and rebase if needed. If this is documentation for a new feature, please add a link to that PR in your description.

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[Copilot]

Pull request overview

Adds the initial Home Assistant documentation page for the new Hot Spring integration, describing setup via the UI, required hardware, supported entities, basic troubleshooting, and removal steps.

Changes:

• Adds a new integration documentation page for hotspring.
• Documents prerequisites and the config flow host field for connecting to the local adapter.
• Describes supported water heater functionality, polling behavior, troubleshooting, and removal.