[Doc] Update README links, mark external links (#18635)

Signed-off-by: DarkLight1337 <tlleungac@connect.ust.hk>

README.md

 <p align="center">
   <picture>
-    <source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/vllm-project/vllm/main/docs/source/assets/logos/vllm-logo-text-dark.png">
-    <img alt="vLLM" src="https://raw.githubusercontent.com/vllm-project/vllm/main/docs/source/assets/logos/vllm-logo-text-light.png" width=55%>
+    <source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/vllm-project/vllm/main/docs/assets/logos/vllm-logo-text-dark.png">
+    <img alt="vLLM" src="https://raw.githubusercontent.com/vllm-project/vllm/main/docs/assets/logos/vllm-logo-text-light.png" width=55%>
   </picture>
 </p>
 
@@ -107,7 +107,7 @@ Please check out [Contributing to vLLM](https://docs.vllm.ai/en/stable/contribut
 vLLM is a community project. Our compute resources for development and testing are supported by the following organizations. Thank you for your support!
 
 <!-- Note: Please sort them in alphabetical order. -->
-<!-- Note: Please keep these consistent with docs/source/community/sponsors.md -->
+<!-- Note: Please keep these consistent with docs/community/sponsors.md -->
 Cash Donations:
 - a16z
 - Dropbox

docs/mkdocs/stylesheets/extra.css

+/* Warning for latest docs */
 .md-banner {
     background-color: var(--md-warning-bg-color);
     color: var(--md-warning-fg-color);
 }
+
+/* https://christianoliff.com/blog/styling-external-links-with-an-icon-in-css/ */
+a:not(:has(svg)):not(.md-icon) {
+    align-items: center;
+
+    &[href^="//"]::after,
+    &[href^="http://"]::after,
+    &[href^="https://"]::after {
+        content: "";
+        width: 12px;
+        height: 12px;
+        margin-left: 4px;
+        background-image: url("data:image/svg+xml,%3Csvg xmlns='http://www.w3.org/2000/svg' width='16' height='16' stroke='gray' viewBox='0 0 16 16'%3E%3Cpath fill-rule='evenodd' d='M8.636 3.5a.5.5 0 0 0-.5-.5H1.5A1.5 1.5 0 0 0 0 4.5v10A1.5 1.5 0 0 0 1.5 16h10a1.5 1.5 0 0 0 1.5-1.5V7.864a.5.5 0 0 0-1 0V14.5a.5.5 0 0 1-.5.5h-10a.5.5 0 0 1-.5-.5v-10a.5.5 0 0 1 .5-.5h6.636a.5.5 0 0 0 .5-.5z'/%3E%3Cpath fill-rule='evenodd' d='M16 .5a.5.5 0 0 0-.5-.5h-5a.5.5 0 0 0 0 1h3.793L6.146 9.146a.5.5 0 1 0 .708.708L15 1.707V5.5a.5.5 0 0 0 1 0v-5z'/%3E%3C/svg%3E");
+        background-position: center;
+        background-repeat: no-repeat;
+        background-size: contain;
+        display: inline-block;
+    }
+}