The outlet my washing machine should be plugged into at the moment does not work. I need to do laundry, so what gauge extension cord can I use to plug it into another, temporary, outlet until an electrician can fix the outlet?

Best to use 3 wire #12 cord.

You should be easily able to find a 6' one sold for window air conditioners.

Run your finger down each row of breakers, lightly. Not enough pressure to flip breakers off, just slightly, so it may expose tripped breaker.
If identified correctly, Turn Washer breaker all the way off, and back on.
Can't go wrong with 12-3