Can I use a standard household extension cord with a submersible well pump?

No, you cannot. Well pumps draw a big startup surge — 2–3 times running amps — so a cheap 16-gauge cord causes voltage drop, overheating, and possible motor burnout. Use at least 12-gauge for a 15-amp pump, or 10-gauge if over 25 feet. Must be three-wire, outdoor rated, and as short as possible. Extension cords are for temporary testing only; permanent installations should be hardwired per code.