[task] Add app-server auth and repository status methods

[phongndo]

Goal

Expose GitHub authentication status and current repository identity through the app-server.

Why this matters

The desktop app needs to explain whether Nexgit can read GitHub data before showing empty dashboards. The TUI needs the same state for first-run and no-auth flows.

Command/API target

Add behavior for these app-server methods:

github.auth.status
github.repo.current

Depends on

• [task] Add GitHub API adapter using gh api #4 for the gh api adapter.
• [task] Resolve current GitHub repository #5 for current GitHub repository resolution.
• [task] Define shared GitHub workflow data types #6 for shared workflow data types.

Out of scope

• Do not launch OAuth or run gh auth login interactively.
• Do not list issues or PRs here.
• Do not update desktop UI in this issue.

Definition of done

• App-server can report whether GitHub auth appears usable.
• App-server can report the current GitHub repository or a clear no-repo state.
• Errors are stable enough for UI loading/error states.

Verification

• cargo fmt --all --check
• cargo check --workspace
• Manual stdio app-server request for both methods included in the PR description.

Contributor notes

Prefer clear errors over clever recovery. The UI can guide users once the state is explicit.