feat(cases): refresh the case picker and add search to Manage (#483)

The Run bar's case picker only loaded /api/cases at page load, so folders
deleted or created on disk stayed listed until a reload. It now refetches
on open and every 5 seconds while open, repainting only when the list
changed and falling back to another case if the selected one was removed.

The Manage tab of Add Case gains a search box filtering by name or path.
Reorder arrows are disabled while a filter is active so a swap cannot
involve a hidden case.


Claude-Session: https://claude.ai/code/session_01GuHtuPiHXdykq9T6rKQJ9n

Co-authored-by: Claude Sonnet 5 <noreply@anthropic.com>

src/web/public/index.html

@@ -3175,6 +3175,7 @@
               <h2>Manage</h2>
             </div>
             <p class="set-section-blurb">Reorder or remove cases, and pick up anything exported from a docker case.</p>
+            <input type="search" id="caseManageSearch" class="set-input" placeholder="Search cases by name or path" autocomplete="off" spellcheck="false" aria-label="Search cases" oninput="app.setCaseManageFilter(this.value)" style="margin-bottom: 8px;">
             <div class="case-manage-list" id="caseManageList">
               <!-- Populated by JS -->
             </div>