You can splice a cut extension cord with a twist and tape.

Yes, but only as a temporary measure until you can replace the cord.

Cut the cord cleanly on both sides of the damage. Strip about a half-inch of insulation off each wire (black, white, and green or bare copper). Twist the matching colors together — black to black, white to white, ground to ground. Wrap each individual splice tightly with electrical tape, then wrap the whole