feat(desktop): polish branch search and session spacing

[usrnk1]

Issue for this PR

No linked issue.

Type of change

• Bug fix
• New feature
• Refactor / code improvement
• Documentation

What does this PR do?

• Focuses branch search after the menu's deferred autofocus, so typing works immediately on mouse and keyboard open.
• Removes the empty session-header slot when the sidebar toggle moves into the open side panel.
• Reduces the gap between patch file icons and paths from 20px to 12px.

How did you verify your code works?

• Five Playwright regressions passed: branch search/filtering/clearing/reopening, LTR and RTL session headers, and desktop/mobile patch rows.
• App, E2E, and session-ui typechecks passed, along with the repository pre-push typechecks.
• Captured matching before/after states using the real UI with fixture data.

Screenshots / recordings

Branch search — opening the menu and typing feature

Before	After

Open context panel — header slot and patch icon spacing

Before	After

Checklist

• I have tested my changes locally
• I have not included unrelated changes in this PR