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/0007_tip_splitting.sql

@@ -0,0 +1,25 @@
+-- Add bather/assistant staff tracking to appointments and tip split ledger (closes groombook/groombook#12)
+
+-- Secondary staff member (e.g., bather) who assisted the primary groomer
+ALTER TABLE "appointments"
+  ADD COLUMN "bather_staff_id" uuid REFERENCES "public"."staff"("id") ON DELETE set null ON UPDATE no action;
+--> statement-breakpoint
+
+-- Stores per-staff tip allocations calculated when an invoice is paid
+CREATE TABLE "invoice_tip_splits" (
+  "id" uuid PRIMARY KEY DEFAULT gen_random_uuid() NOT NULL,
+  "invoice_id" uuid NOT NULL,
+  "staff_id" uuid,
+  "staff_name" text NOT NULL,
+  "share_pct" numeric(5, 2) NOT NULL,
+  "share_cents" integer NOT NULL,
+  "created_at" timestamp DEFAULT now() NOT NULL
+);
+--> statement-breakpoint
+ALTER TABLE "invoice_tip_splits"
+  ADD CONSTRAINT "invoice_tip_splits_invoice_id_invoices_id_fk"
+    FOREIGN KEY ("invoice_id") REFERENCES "public"."invoices"("id") ON DELETE cascade ON UPDATE no action;
+--> statement-breakpoint
+ALTER TABLE "invoice_tip_splits"
+  ADD CONSTRAINT "invoice_tip_splits_staff_id_staff_id_fk"
+    FOREIGN KEY ("staff_id") REFERENCES "public"."staff"("id") ON DELETE set null ON UPDATE no action;