@pekingopera Try this 64-bit portable version of Sumatra PDF to see if it shows all the PDF. If so, that might be a good choice for you as you can select text in the PDF in Sumatra.
As for Adobe, I think everyone looked into this before and there's no fix except for trying to modify the source of the PDF. But, I don't remember if that even worked correctly.
But, you can hit ctrl + p on the page and use the "save to pdf" or the Microsoft print to pdf option. Those will split things up into pages as said. In the print dialog, there's a "more options" section that you can expand to control whether headers, footers, and background graphics are shown. Then, Adobe should have no problem. The content in the PDF just might look more like text than a screenshot like the right-click "save to pdf" option. But, if all you're doing is selecting text from it, that might be okay.