I'm trying to accomplish the delete without using the keyboard (otherwise I would just use the Delete key directly). Highlighting text with the mouse and then right-clicking normally brings up "Delete" as one of many options with most Windows applications (including other browsers, and apps as basic as Notepad and Windows Explorer - see image below for Explorer example), so it's something I've just been doing for many years.
To me this just seems to be a common function, but as I mentioned in my opening post, I have a similar issue with OpenOffice (which is even more surprising, since one of its modules is a word processor). So that made me think that the right-click "Delete" option may be less common for operating systems other than Windows (otherwise, why not include it automatically in an application?).
Well others have said that it's currently not built in to Vivaldi, so I'll probably post it as a feature request for future versions.
Thanks to all who took the time to respond!