You are likely overdriving the speaker and the thermistor is cutting out (as it should). It is there to protect the voice coil. Modern amplifiers are way stronger than the 25W, 35W systems many of these speakers were designed to handle. Back in the day, most of the audio people I know wouldn't play their systems above 50% volume because of distortion and risk of damaging speakers - even though speakers were rated at the same rms wattage as the stereo. The problem, in theory, that distorted (non-sine wave) signals can go well beyond rated rms power.
Yes, bridge with a wire. You'll have to solder it for best results.
your amp will blow these speakers long before it will be damaged. Also, modern amps have protection circuitry to protect themselves.