Keep fan mode in sync with speed and Auto Profile commands

[EdenNelson]

Keep Home Assistant fan speed, fan preset/mode, and Auto Profile commands synchronized with the purifier's active fan mode.

Fixes #47

Summary

• Direct fan speed changes now imply Manual fan mode when no explicit fan preset/mode is requested.
• Local fan preset state updates to Manual immediately when a direct speed change implies Manual mode.
• Auto Profile commands now send the selected profile command and then enter fan Auto mode on models that support fan Auto.
• Core200S is excluded from the follow-up fan Auto command because it does not support fan Auto mode.
• Document that Auto Mode configures automatic behavior and is distinct from the active fan preset/mode.

Validation

• git diff --check passed.
• Editor diagnostics reported no errors in the changed files.
• ESPHome Core400S local compile passed.
• Core400S live testing was performed from EdenNelson/levoit@integration/core400s-auto-profile-stack, which stacked this branch with the active Core room-size round-trip test branch:
  • Home Assistant and the ESP32 Web UI stayed responsive and synchronized.
  • Direct fan speed and Auto Profile interactions behaved coherently with the expected state flow.
  • The remaining ESPHome Web UI limitation is display/control of fan preset mode itself, which is intentionally left to the separate Fan Operating Mode select enhancement.

Scope

This PR keeps fan command mechanics focused on synchronized Manual/Auto behavior. It does not add a Fan Operating Mode select entity, remembered Room Size persistence, Core room-size encoding changes, diagnostics, raw sensors, generated YAML churn, or unrelated model changes.

Disclosure

This PR was prepared with GitHub Copilot/AI assistance and human-in-the-loop review. Live validation was performed by a human on a Core400S using air-purifier-office.yaml.