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/0006_pet_profile_attributes.sql

@@ -0,0 +1,30 @@
+-- Extend pet profiles with grooming-specific attributes (closes groombook/groombook#13)
+ALTER TABLE "pets"
+  ADD COLUMN "cut_style" text,
+  ADD COLUMN "shampoo_preference" text,
+  ADD COLUMN "special_care_notes" text,
+  ADD COLUMN "custom_fields" jsonb DEFAULT '{}' NOT NULL;
+--> statement-breakpoint
+CREATE TABLE "grooming_visit_logs" (
+  "id" uuid PRIMARY KEY DEFAULT gen_random_uuid() NOT NULL,
+  "pet_id" uuid NOT NULL,
+  "appointment_id" uuid,
+  "staff_id" uuid,
+  "cut_style" text,
+  "products_used" text,
+  "notes" text,
+  "groomed_at" timestamp NOT NULL DEFAULT now(),
+  "created_at" timestamp DEFAULT now() NOT NULL
+);
+--> statement-breakpoint
+ALTER TABLE "grooming_visit_logs"
+  ADD CONSTRAINT "grooming_visit_logs_pet_id_pets_id_fk"
+    FOREIGN KEY ("pet_id") REFERENCES "public"."pets"("id") ON DELETE cascade ON UPDATE no action;
+--> statement-breakpoint
+ALTER TABLE "grooming_visit_logs"
+  ADD CONSTRAINT "grooming_visit_logs_appointment_id_appointments_id_fk"
+    FOREIGN KEY ("appointment_id") REFERENCES "public"."appointments"("id") ON DELETE set null ON UPDATE no action;
+--> statement-breakpoint
+ALTER TABLE "grooming_visit_logs"
+  ADD CONSTRAINT "grooming_visit_logs_staff_id_staff_id_fk"
+    FOREIGN KEY ("staff_id") REFERENCES "public"."staff"("id") ON DELETE set null ON UPDATE no action;