chore(mobile): add housekeeping and validation docs

[Shubbu03]

What this does

• adds a CI workflow that installs the workspace, runs the repo typecheck entrypoint, and executes backend tests
• expands the root typecheck script to cover backend, frontend, and mobile TypeScript checks
• adds a dedicated mobile README with emulator, device, APK build, and env setup instructions
• updates the root README to point contributors at the mobile-specific runbook