Some info that might shed light on goofy electronics:
Capacitors!
Spoiler
Capacitors hold a charge, that's what they do. Old TVs (thick ones) produce an image with a cathode ray tube, a device that requires kilovolts (If you charge one of these suckers up, and manage to discharge it through yourself you will be at risk of death if it discharges through your heart) of charge for operation. To provide a clean, smooth voltage to the tube, one or more capacitors would be placed such that if the incoming voltage to the tube dropped for any reason, the stored charge on the capacitor would pick up the slack and keep the signal smooth. This is a basic element of all power supplies.
Typically, to prevent capacitors from continuing to power the device in question and for safety reasons, there's an alternate discharge path in the circuit like a resistor (That Gamecube that had a power light that stayed on, for example). So if the power is unplugged (the main circuit is opened), there's still a closed loop for the charge to drain from the capacitor. If the circuitry in the device was poorly designed or broken, it can take a remarkably long time for those capacitors to discharge. Walking by and providing a static shock (Fun fact: a static shock from your fingers can reach 20kV pretty easily, but with a very low total power) to a 20 year-old switch could break the switch, shorting it together and finally providing a path for the capacitors to discharge.
I can't say this is an exact reason for an old tube TV to start running without a wall connection (no idea what kind of stored power the caps have or what the TV needs). For something simple like a power light staying on, it's a strong explanation.
As for the talking doll, it may have been cheaper (or the designers may have been retarded) to use volatile memory to store the audio information, which would require a small internal battery to keep the information stored. A short could have provided power from that to the speakers. The battery wouldn't be able to power the speakers very long or very loudly, but maybe long enough to scare the unholy pomegranate out of everyone around it. Again, a limited and admittedly mediocre explanation, but it's what comes to mind with my low-level of experience.
Seriously, talking dolls and stuffed animals were a terrible idea and are an unholy kind of creepy. ;_;
So remember kids, if you're ever going to void your warranty on your electronics, properly discharge the device first!
And at the end of the day, even senior electronics engineers chalk up hard-to-explain and unexpected failures to magic smoke!
So don't let the smoke out of your circuits and everything will be okay!
