Use Stamp in actions

You would place the stamp via the Doc.addAnnot() method. See it's documentation here: Acrobat DC SDK Documentation​ There are some samples on that page, but not specifically for stamps. 

This is a simple snippet to add add the "Sign Here" stamp:

var annot = this.addAnnot({
      type: "Stamp",
      page: 0,
      rect: [400, 400, 550, 500],
      AP: "SHSignHere",
  });

To find out what the "AP" of the stamp you want to add is, use the instructions from here: https://acrobatusers.com/tutorials/dynamic_stamp_secrets

Yes. You can add a stamp using a script. You need to know its unique AP value and then you'll be able to add it using the addAnnot method of the Document object.