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/0013_appointment_confirmation.sql

@@ -0,0 +1,7 @@
+ALTER TABLE appointments
+  ADD COLUMN confirmation_status TEXT NOT NULL DEFAULT 'pending',
+  ADD COLUMN confirmed_at TIMESTAMPTZ,
+  ADD COLUMN cancelled_at TIMESTAMPTZ,
+  ADD COLUMN confirmation_token TEXT UNIQUE;
+
+CREATE INDEX idx_appointments_confirmation_token ON appointments (confirmation_token) WHERE confirmation_token IS NOT NULL;