refactor: inline packages/db and packages/types into api package

Phase 2 extraction: groombook/api from groombook/app monorepo.

Changes:
- Move packages/db content to apps/api/src/db/
- Move packages/types content to apps/api/src/types/
- Inline database schema and migrations into api package
- Update Dockerfile to build single package
- Update CI workflow for single-package structure
- Fix vitest.config.ts aliases

Co-Authored-By: Paperclip <noreply@paperclip.ing>

apps/api/src/__tests__/search.test.ts

@@ -23,7 +23,7 @@ const PET_ROW = {
 let clientResults: typeof ACTIVE_CLIENT[] = [];
 let petResults: typeof PET_ROW[] = [];
 
-vi.mock("./packages/db", () => {
+vi.mock("./db", () => {
   // Proxy objects for table/column references — values don't matter for tests
   const tableProxy = (name: string) =>
     new Proxy(