PlayonLinux and Wine generally

I have newly downloaded and installed Zorin 12.4 on amd64

I tried to load multiple Windows programs of multiple types - Adobe Reader DC being the one I want most, though others would also been nice - using both generic Wine (right click on installer file and select open with Wine) and PlayOnLinux, and both downloading the installers myself and letting PlayOnLinux download them. Nothing succeeded in any way for any program except notepad++....which seems to already have a regular linux version, and in any case is of no use to me.

I tried to install multiple programs with PlayOnLinux from its list of officially supported programs, but it gave me an error message every time that it cannot complete because Wine 3.0.4 is not available. When I go to the manage wine versions subsystem in PlayOnLinux it lists only up to 3.0.3 then skips to 3.1 and above. I had it install a higher version like 3.2, but it still said it wants 3.0.4. When I try to submit an error message to PlayOnLinux it says I cannot because I do not have the most recent version of PlayOnLinux and I should look at their website. When I follow the instructions on their website for installing the latest PlayOnLinux version 4.3.4 for the Ubuntu X on which I understand Zorin 12.4 is based, it all processes in terminal but then says that I already have the latest version of PlayOnLinux, something like "4.2.12-1zorin2"

The upshot seems to be that the only version of PlayOnLinux Zorin will allow me to use is out of date?

Hi and welcome! The reason why you won't be able to install a different version is the one in the software channel will have been made to match the kernel(s) currently in use by Zorin or some other library dependencies that would prevent the one from PlayOnLinux installng.
Remove any installation of PlayOnLinux and WINE, then install Synaptic Package Manager - search for WINE and you will see that the latest version is 4.0:

Please be aware, even though you install the latest version, not all Windows applications may work - this is not a WINE issue but the way some Windows programs have been written. It may be looking for a Windows Certificate to tell it that it is installing to a Windows machine. There are plenty of pdf viewers in GNU/Linux for you to use - you don't need to be reliant on Adobe Acrobat DC.