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/style.css

@@ -1174,6 +1174,33 @@ body {
   color: var(--text-dim);
 }
 
+/* ============================================================
+   Sidebar sticky footer (+ New button)
+   ============================================================ */
+
+.sidebar-footer {
+  padding: 8px;
+  border-top: 1px solid var(--border-subtle);
+  flex-shrink: 0;
+}
+
+.sidebar-new-btn {
+  width: 100%;
+  background: none;
+  border: 1px dashed var(--border);
+  border-radius: 4px;
+  color: var(--text-dim);
+  font-size: 12px;
+  font-family: var(--font-ui);
+  padding: 8px;
+  cursor: pointer;
+}
+
+.sidebar-new-btn:hover {
+  color: var(--text-muted);
+  border-color: var(--text-muted);
+}
+
 /* ============================================================
    Responsive overlay sidebar at <960px
    ============================================================ */