Opening a PDF brings up the Print dialog

The print permission is in the pdf file itself and is mostly used by Adobe products. Some products pay it attention, and some others print it anyway.
It is quite easy to remove.

See the article How To Unlock Adobe PDF Files, where it explains how to use Freeware PDF Unlocker to remove this and other passwords.

Warning : Comment by Chris Betti says

As of Oct 7, 2013, the Freeware PDF Unlocker setup installs unwanted software without confirmation, and fails to install the product itself.

Use older versions (if can be found) or search for another product.

When you choose File > Print..., there's a button labeled Subset: All pages in range. You can click on that button and Odd Pages Only or Even Pages Only. Do one of those and print the document. Flip the pages over and put them back into the input tray and print the document again after choosing the other. Practice on a short document first to make sure you know which way the pages need to be collated after the first print.

How many pages you're able to print depends on whether you have the extra large or regular black cartridge, whether you print in color or set the output to black only, what print quality you choose (draft, etc.) and how much you've used the cartridge(s) before. The density of text and the amount of graphics will have a lot to do with it as well. HP rates the regular black cartridge for that printer at 200 pages and the extra large black cartridge at 700 pages.