Import PDF to QGIS

It should not be too hard to achieve this by converting the PDF to an image format (say Tiff) and georeferencing that image and projecting it in the same projection as the data you are overlaying. You will not be 100% accurate but with care you can get a good result.

A few random thoughts:

• As seems typical with so many maps, there are no graticules or tic marks to help you but you could get reference points from Tiger data or maybe even your KMZ.
• When you convert the image make sure it is NOT saved in a progressive format, especially if you are using GIMP (where this is the default). See my answer in this thread.
• If you are having trouble opening the KMZ in QGIS, you can open it in Google, save it as a KML and that will open fine.
• To georeference the image go Raster->Georeferencer
• It looks like the image has been exported from a GIS rather than scanned from paper, which is a good thing as there will be fewer inaccuracies from paper shrinkage, folding etc.
• I would convert the KMZ file to a shapefile first and then edit the Shapefile's attributes, just to be sure of preserving the original data, although you can always download it again in this case.

In the 3.10 version, you don't need to convert the PDF to another format. I have just imported it as a new raster layer. It works.

Afterward, you must georeference this image, as per the answer above.