feat(web): move the chat action cluster to the pinned bottom and align the button sets

- task 01: relocate the .chat-actions row (New chat + Share, comments byte-identical with a Phase 65 note) from the top of the column to the bottom of .chat-shell, directly above the composer
- task 02 (owner-locked A1): wrap the row + #composer in ONE sticky .chat-bottom unit (position: sticky; bottom: env(safe-area-inset-bottom, 0), no z-index) — the pills stay at the bottom of the screen at every scroll position and settle into flow above the footer
- task 03 (owner-locked A2): right-align the bottom row to the column's right edge (justify-content: flex-end), mirroring the right-aligned Save-as-doc corner; the five action pills share one 44px / 999px-pill geometry
- task 04: dedicated Playwright suite tests/e2e/test_bottom_chat_actions.py (resting geometry, the A1 pin across the sticky range, A2 alignment + DOM order + mobile stack + 360px overflow bound + 44px touch targets, New chat / Share click-through) — green in isolation
- task 05: regression matrix green in isolation (pinned_composer 4, save_share_ux 5, chat_persistence 4, share_chat 4, chat_history 5, smoke 3); full gate green — unit + integration pass, app/ coverage 99% (>90%), ruff + pyright clean

tests/unit/test_chat_persistence.py

@@ -178,8 +178,11 @@ def test_new_chat_button_lives_only_on_the_chat_page() -> None:
     """#new-chat-btn is a real type=button with an accessible name. Moved
     from the shared header bar to the chat page at owner request
     (2026-08-28): it lives ONLY in index.html — inside <main>, in
-    .chat-shell, above the #messages section (the single module binding
-    + the no-op guard are pinned in test_shared_header.py)."""
+    .chat-shell. Phase 65 (2026-09-01, `TODO.md` L3, owner confirmation)
+    relocated the .chat-actions row from the top of the column to the
+    bottom: the button now sits BELOW the #messages section, directly
+    above the composer (the single module binding + the no-op guard are
+    pinned in test_shared_header.py)."""
     html = _index()
     btn = re.search(r'<button[^>]*id="new-chat-btn"[^>]*>', html)
     assert btn, "index.html must contain #new-chat-btn"
@@ -191,8 +194,10 @@ def test_new_chat_button_lives_only_on_the_chat_page() -> None:
     shell_idx = html.find('class="container chat-shell"')
     assert shell_idx != -1 and shell_idx < btn.start(), "the button belongs in .chat-shell"
     messages_idx = html.find('id="messages"')
-    assert messages_idx != -1 and btn.start() < messages_idx, (
-        "the button sits above the #messages section"
+    messages_end = html.find("</section>", messages_idx)
+    composer_idx = html.find('<form class="composer" id="composer"')
+    assert -1 < messages_idx < messages_end < btn.start() < composer_idx, (
+        "the button sits below the #messages section, above the composer"
     )
     # No other page carries the button anymore (owner request 2026-08-28).
     for other in (SOURCES_HTML, GIT_SOURCES_HTML, DOCUMENT_HTML, LOGIN_HTML, TUNING_HTML):