Can I paste my Fathom document into another program?

You can't paste the entire document, but you can paste pictures of individual objects. Click on an object to select it and choose Copy Picture from the Edit menu; then switch to the other program and paste the picture.

Note that you can only paste one object's picture at a time. Also note that the objects won't update automatically if you change the original document; you would have to copy and paste them again.