Hi,
Thank you for the details, it helps a lot!
I have logged in to check, and there is indeed something missing as WordPress itself should automatically generate covers for PDF documents during upload. The most likely cause is that ghostscript is not installed, it is not well documented, but WordPress needs that to generate thumbnails.
If those thumbnails are not present, Ajax Search Pro will try alternative methods via ImageMagick, but the extension has to have access to PDF documents. It can be enabled via the policy file. Without that the script has no access to the PDF files. Correcting the policy file will very likely resolve the issue.