Add local development build workflow

- Add Makefile targets: install, type-check, lint, build-local
- Add type-check script to backend/package.json
- Create backend/.gitignore for build artifacts

Enables quick TypeScript error detection without Docker rebuilds.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

backend/.gitignore

@@ -0,0 +1,22 @@
+# Dependencies
+node_modules/
+
+# Build output
+dist/
+
+# Test coverage
+coverage/
+.nyc_output/
+
+# Logs
+*.log
+npm-debug.log*
+
+# Environment
+.env.local
+.env.*.local
+
+# Editor
+.DS_Store
+*.swp
+*.swo