High temperatures are usually the problem for capacitors. Lower temperatures usually are not. Higher temperatures can cause aging or drying out of the capacitor. You got your money's worth. Under normal operating conditions electrolytic caps can go longer than 15 years. High temps (over 212F, 100C) will reduce the life time. Your new cap will serve you a long time.
The crud on the capacitor looks like some of the blue smoke tried to escape and got condensed
Anyway, it appears you've zeroed in on the problem (in a timely manner, no damage to the motor), and with a new capacitor you should be good to go. As far as I know, the electrolyte in a capacitor is a pretty heavily loaded solution. So it shouldn't freeze until you get further below zero than even the deepest winter.
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