How do I cut-and-paste plain text from Notes to another iOS app?
To get rid of the formatting, copy the data to a coding app like iEditor from iEditor copy the data and paste to where you need it. 
https://itunes.apple.com/us/app/ieditor-text-code-editor/id562162723?mt=8

To avoid the double copy, work in iEditor. I do this all the time on macOS. 

Copy and paste involves how the "copy" app and the "paste" app handling the text. The copy app may make several formats of the data available. One format must be a styled version. Another will likely be a text only version.  The "paste" app is free to select any of the formats. 

What I do on macos, when the "copy" app is passing formatted data that I do not want, is to copy the data to a coding editor.  The coding edit will pick the data form without any styling information.  You end up with plan text.  I then copy and paste into the destination app.