Thanks Sam. I'm actually trying to copy/paste the features from one feature class into an existing feature class, not create a new one. This was straight forward in ArcMap: start editing, select, ctrl+C, ctrl+V and done.
This is what happens when I click on a feature to select it in the attribute table. The drop down arrow for paste special is grayed out. When I click Paste, it copies the whole layer, not the selected features.

It works fine selecting them with the Selection tool so I'm not sure why selecting features another way is not working.