feat(app): share Home management actions with Side bar - #2

Issue for this PR

Targets your sidebar-mode branch so this can be folded into anomalyco#60867. Related: anomalyco#48882.

Type of change

• New feature
• Refactor / code improvement

What does this PR do?

Adds Home's project menu to Side bar: new/import session, edit project (including its name), reveal folder, clear notifications, and close. The menu and actions are shared with Home. Right-click and keyboard access work too.

Closing a project also hides its unsent drafts while retaining history. Session rows gain Home's Export… action. Project counters no longer intercept the header's pointer area.

A separate current-dev integration branch resolves the two Inbox conflicts against dev@156bb6101; this PR keeps those upstream changes out of the feature diff.

How did you verify your code works?

• bun run check: 35/35 packages on both branches; official app TypeScript check and changed-file lint pass. Five production sidebar/Inbox cases also pass on the current-dev integration.
• Production Chromium: all 78 Home/tab keeper cases verified; the two management flows also pass 20 executions with five workers and no retries. Windows fixture shortcuts were normalized locally on macOS; native-paste cases were retested with the native shortcut. These adaptations are not committed.
• Home → session navigation: 211 ms stable content before; 172 ms median over three runs after (local diagnostics, no performance threshold). The other two navigation benchmark cases already fail on the unmodified base.
• typecheck:e2e reports the same 13 diagnostics as 703db47378, with byte-identical output.

Visual

<svg xmlns="http://www.w3.org/2000/svg" width="240" height="72" viewBox="0 0 240 72" fill="currentColor"><style>.sidebar-demo-menu{animation:sidebar-demo-show 3s ease-in-out infinite}@keyframes sidebar-demo-show{0%,100%{opacity:.45}40%,80%{opacity:1}}@media(prefers-reduced-motion:reduce){.sidebar-demo-menu{animation:none}}</style><rect x="8" y="6" width="94" height="60" rx="4" fill="none" stroke="currentColor" opacity=".35"/><text x="15" y="21" font-size="9">Projects</text><path d="M18 34h60M24 46h49M24 57h38" stroke="currentColor" stroke-width="4" opacity=".35"/><circle cx="93" cy="20" r="1"/><circle cx="93" cy="24" r="1"/><circle cx="93" cy="28" r="1"/><g class="sidebar-demo-menu"><rect x="112" y="11" width="119" height="50" rx="4" fill="none" stroke="currentColor" opacity=".4"/><text x="120" y="25" font-size="8">New session</text><rect x="116" y="29" width="111" height="13" rx="2" opacity=".12"/><text x="120" y="39" font-size="8">Edit project</text><text x="120" y="53" font-size="8" opacity=".7">Close</text></g></svg>

Screenshots / recordings

Before	After

Checklist

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