From a PDF, is there a way to figure out what settings were used to export the file from InDesign (High Quality Print/Press/Smallest File Size, compression settings, etc.)?
With Acrobat you can check compliance with things like PDF/X and you can check the originating application, image resolutions, etc, but I don't believe you can check what the job options were specifically titled. Especially since job option titles mean nothing really. You can retitle "High Quality Print" or any of them to anything you want.
View > Tools > Print Production > Preflight > Options > Browse Internal PDF Structure then click the Browse Internal Document Structure Button (top right of the pane) and expand the Document Info list....
This will tell you if a PDF conforms to standards such as PDF/X and it will list the original application which created the PDF.
You can also expand the Document Root list item to dig into specific areas. And the Preflight window in general will perform checks for certain compliance settings such as color space, resolution, etc.