Initial extraction: groombook/api from groombook/app monorepo

Part of GRO-802 monorepo breakdown.

Changes:
- Extract apps/api/ as the main API service
- Inline packages/db/ (database schema, migrations, utilities)
- Inline packages/types/ (shared TypeScript types)
- Add CI workflow for lint, typecheck, test, build, docker
- Port Dockerfile with 4 stages: runner, migrate, seed, reset

Co-Authored-By: Paperclip <noreply@paperclip.ing>

packages/types/tsconfig.json

@@ -0,0 +1,13 @@
+{
+  "compilerOptions": {
+    "target": "ES2022",
+    "module": "NodeNext",
+    "moduleResolution": "NodeNext",
+    "strict": true,
+    "noUncheckedIndexedAccess": true,
+    "skipLibCheck": true,
+    "outDir": "./dist",
+    "rootDir": "./src"
+  },
+  "include": ["src"]
+}