Persist room solve history and past sessions

Comienza revisando PR #171, PR #140, #174 y #175 para entender el trabajo de persistencia, retención, read API y paridad ya planificado. Resuelve el comportamiento documentado de retención y privacidad y, después, sigue la secuencia de implementación hasta la profile/session-history UI. Se considera completado cuando los intentos finalizados se pueden consultar con paginación estable, los datos privados están protegidos y el comportamiento del rollback está documentado.

Users want to revisit solves and sessions after the live room is gone.

Feedback source:

• Would it be possible to access past solves, like up to 30 days?
• You could make it so you can look at all your past solves.
• Scrambles should remain available after completion while respecting room retention.
• Audio/video feedback also mentioned an option to save the session.

Current foundation

• PR #171 adds PostgreSQL dual writes for durable solve results while MongoDB remains the live read source.
• PR #140 prevents normal-room TTL deletion by hiding expired rooms, but it predates the current dev branch and must be rebased/redesigned.
• #174 defines the first feature-flagged solve-history read API.
• #175 covers the historical backfill and parity checks required before that read path can be trusted.

Implementation sequence

• Decide and document retention and privacy behavior for public, private, and deleted/hidden rooms.
• Rework or split PR #140 on top of the modernized dev branch.
• Deploy and stabilize PostgreSQL dual writes.
• Complete #175 and verify historical/ongoing parity.
• Complete #174 with authenticated cursor pagination.
• Add a profile/session-history UI.
• Reuse the read API for previous scramble review (#161) and csTimer export (#163).

Acceptance criteria

• Completed attempts persist with event, time, penalties, scramble, timestamps, and stable user/session references.
• Users can browse their history after the live room is hidden or expired.
• Private-room and deletion/retention expectations are enforced server-side.
• History pagination is stable and does not expose other users' private data.
• The migration has a documented disable/rollback path while MongoDB remains the live source of truth.