Thanks for your prompt reply, Chuck.
I did some more testing using a clean install of FF34 on another machine, and I was able to easily reproduce the bug.
It turns out to not be random at all. Here's how to reproduce it:
If you right-click on an area of a page that's NOT a text input field and you select the Paste Email context menu item, it will disappear for good. The only way to get it back is to restart Firefox.