How to get Paste menu to show in textarea on Chrome?
As my question indicated, long pressing in a textarea (i.e. the multi line html input box) will highlight the word (or the nearest word/whitespace), and double tapping does what a single tap does: shows the blue cursor button. 

The trick I was missing is that once the blue cursor button is showing, I tap this button to get the Paste menu to show. Once the Paste menu is showing, clicking around will move the cursor and the Paste menu.