When I copy HTML to Google Drive, I lose the URLs in the text. Is that normal behavior? If so, how do I get around it to keep the URLs?
You cannot (and shouldn't) copy raw HTML into a Google Drive document and expect it to render like how you see it in a browser. Instead, you should copy the rendered text (with the clickable links) and paste it directly into the Google Drive document.
The only problem with this, however, is that the formatting of the text will be different from the formatting of the rest of the text in the document.