[Feat] Add experimental Code Mode setting

[roomote-roomote]

​Opened on behalf of @mrubens. Follow up by mentioning @roomote-roomote, in the web UI, or in Slack.

What changed

• Added an admin-only Code Mode toggle under Experimental settings.
• Reused the existing deployment setting and propagated it consistently to coding tasks and Fast sessions.
• Added focused coverage for persistence, authorization, and enabled/disabled runtime propagation.

Why this change was made

Code Mode can reduce the tool definitions sent with each request by deferring eligible tools until they are needed. This makes the capability available as an explicit deployment opt-in.

Impact

Admins can enable Code Mode from Settings > Experimental. The setting controls the targeted runtime capability without enabling unrelated experimental features or changing existing tool permissions.

Screenshots

[roomote-community]

No code issues found. See task

_{Reviewed 7f31f1e}

[roomote-roomote]

Fixed the stale Experimental settings navigation expectation in 7f31f1e. The focused test and all 2,398 web client tests pass locally.

The first fresh CI run then hit an unrelated, non-reproducible DB notification test failure: that exact test passed in isolation, in its full file, and across ten repeated runs. The failed Test job was rerun unchanged on the same head and passed. All latest-head required checks are green, the synchronized Roomote review reports no code issues, and there are no unresolved review threads.