Answer: First of all, the white on the leaves is probably powdery mildew, a common disease on pumpkin plants and nothing to worry about at this point in the season. (If it occurs early in the season, it can reduce yields so you would want to take some control measures.)
It's best to let pumpkins mature fully right on the vine. To test for maturity, press your fingernail into the rind. If the skin doesn't dent, then the shell is hardenend and it's ready to harvest. The one you lost probably rotted because it was in contact with moist soil, which hastens decomposition. In the future, if you want to leave the pumpkins on the vine a bit longer, simply place a piece of wood under each pumpkin, so it's not resting right on the ground.
Your pumpkins should last a few months if you take a few precautions: First, let the newly-harvested pumkins sit in the sun or in a well-ventilated place for about a week. After that, the ideal storage temperature is 50-55 F. However, I've kept pumpkins on my front stoop up to and beyond Halloween by bringing them indoors whenever a frost threatens.
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