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/db/drizzle.config.ts

@@ -0,0 +1,10 @@
+import { defineConfig } from "drizzle-kit";
+
+export default defineConfig({
+  schema: "./src/schema.ts",
+  out: "./migrations",
+  dialect: "postgresql",
+  dbCredentials: {
+    url: process.env.DATABASE_URL!,
+  },
+});