Fix Docker build: compile TypeScript packages before runtime stage

The API Docker image was copying raw .ts source files for @groombook/db
and @groombook/types into the runtime stage, but tsx (the TS loader) is
a devDependency stripped by --prod install. Node.js cannot load .ts files
natively, causing ERR_UNKNOWN_FILE_EXTENSION at startup.

- Add build scripts to db and types packages
- Add outDir to types tsconfig
- Update package.json main fields to point to compiled dist/
- Build packages in dependency order in Dockerfile
- Copy only dist output to runtime stage

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

packages/types/tsconfig.json

@@ -5,7 +5,8 @@
     "moduleResolution": "NodeNext",
     "strict": true,
     "noUncheckedIndexedAccess": true,
-    "skipLibCheck": true
+    "skipLibCheck": true,
+    "outDir": "./dist"
   },
   "include": ["src"]
 }