Extract PDF pages with annotations only

Yes, both of these things are possible. There is a JavaScript function for acquiring annotations on a per page basis, and there are functions for extracting and inserting pages. The idea is to write a script that loops through all the pages in a PDF and acquire the list of annots for each page. If the list is empty then go on to the next one, if it's got something in it, then extract/insert the page into a new PDF. This is an automation script so it will need to be either run from the console window, or you will need to place it in a folder level script file. You can look up both of these topics on this web site.

To do this for a folder of files you'll need to use Batch Processing, which is only available on Acrobat Pro. A Batch sequence is also a great place to put an Automation script.

Soon, there will be an Actions gallery on this site. Actions are the Acrobat X equivalent of Batch Processing. I've created one for the gallery that does exactly what you are asking about. Even though it's for AcrobatX, the script will work in previous versions of Acrobat, and in Standard.