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/migrations/0018_backfill_staff_user_id.sql

@@ -0,0 +1,14 @@
+-- Backfill staff.user_id for staff records created before Better-Auth integration.
+-- Staff records that predate this migration have user_id = NULL; the resolveStaffMiddleware
+-- now falls back to staff.id (dev mode) and oidcSub (production) so these records still work.
+-- This migration populates user_id for the known demo/dev staff seeded by seed.ts.
+
+-- Create demo Better-Auth users for seeded staff (these match the ba-user-* IDs used in tests)
+INSERT INTO "user" (id, name, email, email_verified, created_at, updated_at)
+VALUES ('ba-user-manager', 'Demo Manager', 'demo-manager@groombook.dev', true, NOW(), NOW())
+ON CONFLICT (id) DO NOTHING;
+
+-- Link the demo manager staff record to the Better-Auth user
+UPDATE staff
+SET user_id = 'ba-user-manager', updated_at = NOW()
+WHERE oidc_sub = 'demo-manager-001' AND user_id IS NULL;