Is there any way to get a Google Docs document to include its comments in the PDF file that it creates when you click the "print" button?
It doesn't appear possible at the moment;
Unfortunately you cannot print or export comments to PDF at this time
What is suggested is exporting your comments as a .doc file (then converting to .pdf perhaps ?) which apparently retains the comments.
Hope this helps.
I have tested this on linux machine. the work around is download it as an .odt document. Then open it in libreoffice and then export it as pdf. Since libreoffice is available for windows also so my guess is it will work for windows also.
After exporting it the comments is visible as an icon where you have commented. See screenshot below.
Google docs screenshot
Generated pdf screenshot
Comment in comment box
You could use this Chrome extension to take a multi-page screenshot of the document and print that capture:
You can use the Chrome extension Google document with comments printer by Deemwar to print a Google document with comments.