All native Apple apps that handle documents or photos, including Preview, Pages, Numbers, TextEdit, and other 3rd party apps like PDFpen let you rename and/or move an open document in one easy step in the viewing app as I described in my original post. In PDFelement, you apparently have two options. First, you can do a Save As on the open document. That mimics the ease of Apple's native apps. But it also leaves the original document where it was previously saved as an orphan. Now you have to find and delete it in Finder. Windows works this way, also. The second method in PDFelement is as I described above, right click on the file name in the tab, open it in the Finder and rename and/or move it from there. The good news is that you can do this while the document is open, and the new name will appear in the tab. The bad news is that it requires unnecessary steps that don't take advantage of the native Apple method.
I download many pdfs from the Web for business and research that I have no intention of saving, so by default they all go to my Download folder. I open them from there and then decide what to do with them. Yes, I could do a Save As in my browser, but I don't know which ones to do that to until I read them.
Thanks for forwarding my feature request, Rebecca.