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/0025_rate_limit.sql

@@ -0,0 +1,6 @@
+-- Better-Auth rate limiting table (GRO-574)
+CREATE TABLE "rate_limit" (
+  key TEXT NOT NULL PRIMARY KEY,
+  count INTEGER NOT NULL,
+  last_request BIGINT NOT NULL
+);