style(backend): rustfmt the whole tree; gate cargo fmt --check in CI

The backend had never been run through rustfmt. Doing it in one mechanical pass (134 files)
so no future functional diff is buried under formatting churn, then gating `cargo fmt
--check` in checks.yml so it stays clean.

Formatting only — no logic, SQL, or behaviour changed. Verified after the reformat:
cargo test 56 passed, clippy --all-targets -D warnings clean, cargo fmt --check clean.

This is the deferred cleanup noted when CI's Format step was first left out.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

backend/src/handlers/public.rs

@@ -1,7 +1,7 @@
 //! Unauthenticated, read-only endpoints safe to expose before a user has joined.
 
-use axum::extract::State;
 use axum::Json;
+use axum::extract::State;
 use serde::Serialize;
 
 use crate::state::AppState;