Adopt rustfmt and clean up clippy warnings

Formats the whole backend to rustfmt defaults and fixes all 23
outstanding clippy warnings (collapsible ifs, clone-to-slice-ref,
type-complexity, single-arm match, unnecessary is_none check), then
wires cargo fmt --check and -D warnings into make lint so both stay
enforced going forward.

Co-Authored-By: Claude Sonnet 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01TeyhRfMYjMditek6geDRoQ

Makefile

@@ -23,7 +23,8 @@ test-fe:
 lint: lint-be lint-fe
 
 lint-be:
-	cargo clippy --all-targets
+	cargo fmt -- --check
+	cargo clippy --all-targets -- -D warnings
 
 lint-fe:
 	cd webapp && npm run lint