I'm trying to drag a paragraph style onto the 'create new style' icon in the paragraph styles panel in order to duplicate the style, but the cursor changes to a circle with a slash through it, signifying this cannot be done. I know that a new style can be duplicated in this way, and I'm wondering what might be causing this behavior. (I'm also not able to drag the style onto the trash for the same reason.)
This is exactly what you will see if you try to duplicate the [Normal Paragraph Style] in Illustrator, so I suspect that's what you're doing. (This applies even if you've changed the font, leading, etc. from the defaults.) You can't drag this style to the trash because it is the default paragraph style and can't be deleted.
What you do is this:
Target the paragraph with the style you want to duplicate.
Alt/Opt-Click the New Paragraph Style icon. In the dialog, give the style a name and click OK.
You will be able to duplicate this style by dragging it to the New Paragraph Style icon.