Hello LaPastenague. Thanks for offering to help.
Make and Model: CenturyLink C1100T 802.11n WiFi Router. It may have been built by RoHS. (I have to run DSL up here in the mountains of Colorado.)
I don't know the model of my airport. I don't even know what an airport is. Maybe I don't have this loaded at all as the icon has never appeared until I used the following: https://www.youtube.com/watch?v=IKHlJdyNP-k.
Indeed, I can print docx, pdf, jpgs, tiffs, etc. Some documents that appear in Preview, I have to first save to my desktop (usually as pdfs) and then open them before printing them. It is the web browser print that fails. I use Chrome all of the time as my office computer (a PC) used Chrome (I'm retired now).
I receive the following error message on a greyed out print screen: "The selected printer is not available or not installed correctly. Check your printer or. try selecting another printer." This appears, for instance, when I am trying to print a recipe from the NYTimes Food section, clicking on the print icon on the webpage.
As I wrote, I cannot print from Preview. I had to save the document, usually as a pdf, to my desktop before I can print.
I've not changed anything on the computer except for upgrading to Catalina 10.15.3.
We did install a new router as the old one was needing to be restarted more often than it needed to previously. CenturyLink sold us the router described above, claiming it would also increase our speed; however, we haven't noticed much of an increase in speed. (This could also be the result of our daughter and family who have come to live with us from the Bay Area which has added two more laptops and two more phones to our personal two laptop/two phone WiFi load. The son-in-law is working from our home every weekday.)
I thought the instructions from https://www.supportvideos.ext.hp.com/detail/video/1251249230001/setting-up-an-hp-printer-wirelessly-in-mac-os-x really arcane and difficult to follow, setting up an adhoc network. The first thing the video asked me to do is to use my Airport to connect to the printer and then print the printer's configuration page. I kept thinking something must be an easier fix than that.
Thanks for your time and help.
I'm hoping I can do all this without too much of a problem.
Paul