feat: add sidebar + New sticky footer with inline input

- Add div.sidebar-footer with button#sidebar-new-session-btn.sidebar-new-btn
  ('+ New') to #session-sidebar in index.html, positioned after #sidebar-list
- Add CSS for .sidebar-footer (padding: 8px, border-top, flex-shrink: 0)
- Add CSS for .sidebar-new-btn (width: 100%, dashed border, dim text, hover
  effects with muted text and border color)
- Bind #sidebar-new-session-btn click to showNewSessionInput() in
  bindStaticEventListeners()
- Add 15 tests covering HTML structure, CSS rules, and JS binding

🤖 Co-authored-by: Amplifier <amplifier@example.com>

muxplex/frontend/index.html

@@ -45,6 +45,9 @@
           <button id="sidebar-collapse-btn" class="sidebar-collapse-btn" aria-label="Collapse session list">&#8249;</button>
         </div>
         <div id="sidebar-list" class="sidebar-list"></div>
+        <div class="sidebar-footer">
+          <button id="sidebar-new-session-btn" class="sidebar-new-btn">+ New</button>
+        </div>
       </div>
       <div id="terminal-container" class="terminal-container"></div>
     </div>