feat(tui): UX polish — multi-line input, timestamps, update check & v0.0.3 (#85)

* feat: implement background update check and startup notifications

* feat: enhance user experience with timestamp notifications and UI polish

* feat: implement multi-line chat input with Enter-to-submit and modifier+Enter newline

* update

* update

* v0.0.3

* feat: improve code readability with consistent formatting in TUI and test files

* feat: update PyPI badge version to v0.0.3 in README files

* feat: add docstrings for test classes in test_update_check.py

.github/assets/badge-pypi-dark.svg

@@ -5,5 +5,5 @@
   <rect x="54" y="5" width="62" height="24" rx="6" fill="#1565c0"/>
   <text x="85" y="22" text-anchor="middle"
         font-family="Inter, -apple-system, system-ui, sans-serif"
-        font-size="13" font-weight="700" fill="#ffffff">v0.0.2</text>
+        font-size="13" font-weight="700" fill="#ffffff">v0.0.3</text>
 </svg>

.github/assets/badge-pypi-light.svg

@@ -5,5 +5,5 @@
   <rect x="54" y="5" width="62" height="24" rx="6" fill="#2563eb"/>
   <text x="85" y="22" text-anchor="middle"
         font-family="Inter, -apple-system, system-ui, sans-serif"
-        font-size="13" font-weight="700" fill="#ffffff">v0.0.2</text>
+        font-size="13" font-weight="700" fill="#ffffff">v0.0.3</text>
 </svg>