Thanks for sharing! Interesting feature.
Regarding the “Line Numbers Sometimes” mode. It was not clear to me.
In Settings | Edit Current Profile | Advanced I saw the following window:
The modifiers for URL hints seems unrelated. And what is a field “Shortcut for peeking the primary screen” is not clear to me.
But I found it is working the following way. I selected several modifiers there (Ctrl and Shift), applied changes. Then in Konsole I pressed Ctrl + Shift + \, and line numbers are shown while you hold this combination.
Also note that numbering is not aware of wrapped lines. So when you scale your text or change the window size, and some lines got (un)wrapped, the numbering will be changed. In other words, it is not like in text editor.