Persist room solve history and past sessions

[coder13]

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 Add PostgreSQL dual writes and room metrics #171 adds PostgreSQL dual writes for durable solve results while MongoDB remains the live read source.
• PR Persist solves by hiding expired rooms #140 prevents normal-room TTL deletion by hiding expired rooms, but it predates the current dev branch and must be rebased/redesigned.
• Add a feature-flagged PostgreSQL solve-history read API #174 defines the first feature-flagged solve-history read API.
• Backfill MongoDB data into PostgreSQL and verify ongoing parity #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 Persist solves by hiding expired rooms #140 on top of the modernized dev branch.
• Deploy and stabilize PostgreSQL dual writes.
• Complete Backfill MongoDB data into PostgreSQL and verify ongoing parity #175 and verify historical/ongoing parity.
• Complete Add a feature-flagged PostgreSQL solve-history read API #174 with authenticated cursor pagination.
• Add a profile/session-history UI.
• Reuse the read API for previous scramble review (Show previous scrambles for completed solves #161) and csTimer export (Export room sessions to csTimer #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.