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/sse.rs

@@ -1,14 +1,14 @@
 use std::convert::Infallible;
 use std::time::Duration;
 
+use axum::Json;
 use axum::extract::{Query, State};
 use axum::response::sse::{Event, KeepAlive, Sse};
-use axum::Json;
 use futures::stream::Stream;
 use serde::{Deserialize, Serialize};
-use tokio_stream::wrappers::errors::BroadcastStreamRecvError;
-use tokio_stream::wrappers::BroadcastStream;
 use tokio_stream::StreamExt;
+use tokio_stream::wrappers::BroadcastStream;
+use tokio_stream::wrappers::errors::BroadcastStreamRecvError;
 
 use crate::auth::middleware::AuthUser;
 use crate::error::AppError;
@@ -42,7 +42,10 @@ pub async fn issue_ticket(
     let server_time = sqlx::query_scalar("SELECT NOW()")
         .fetch_one(&state.pool)
         .await?;
-    Ok(Json(StreamTicketResponse { ticket, server_time }))
+    Ok(Json(StreamTicketResponse {
+        ticket,
+        server_time,
+    }))
 }
 
 /// SSE stream endpoint. Authenticates via a single-use ticket (see