That message about AUTH_TYPES can be ignored (it has to do with HPLIP checking for updates which you don't want to do).
I would suggest that you take the printer system back to infancy; open CUPS in a browser and delete any existing printer entries for that printer (you don't configure an H-P printer in CUPS, that is done for you by
hp-setup).
If your printer is a combination printer/scanner, make sure that your user account (and any user accounts that use the printer or scanner) are in the
lp and
scanner groups.
Open a terminal window and execute
hp-setup. You will, as you've discovered, need the root password (userid and password are case-sensitive; it's
root and whatever password).
As you step through
hp-setup, you'll identify the connection (USB? Ethernet?) and
hp-setup will find the printer and identify the printer for you. You want to add a short name for the printer, say, laserjet or something similar (you'll use that name for printing with the command line, for example).
I have an H-P Business Inkjet 2280tn that is connected via Ethernet and an H-P Photosmart C4680 connected via USB both were added with
hp-setup and have worked just fine through a number of upgrades to
hp-setup; it'll work if you let it, eh?
Take the recommended settings (they typically work fine);
hp-setup "knows" the proper pre- and post processing settings for every printer (and other devices) H-P makes, trust it to make the best choice.
HPLIP has been included with Slackware for some years and it kept up-to-date when H-P releases a new version (as you noted, the current version in 14.1 is 3.13.10).
The one last step (and really the only thing you want to do with CUPS is open a browser and go to
http://localhost:631. Click on the printer name (that you added; e.g., laserjet, then click on Administration and Set as Server Default. That way, you can simply
and get a pretty-print manual page without having to define the printer name to
lp.
Hope this helps some.