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>

.dockerignore

@@ -0,0 +1,11 @@
+node_modules
+.git
+*.md
+.github
+apps/e2e
+apps/web/dist
+apps/api/dist
+packages/db/dist
+packages/types/dist
+.turbo
+screenshots/