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Will the fan cool down the print too fast?
With some parts it may cause you troubles, but in a general way, you should specify to the fan to be on only after a few layers in order to prevent warping.
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Would it cool down the extruder too?
Yes, absolutely. To get more precision on the air flow, a support to mount the fan on the extruder, with a fan duct embedded is, to me, the best.
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What if it was insulated?
It will be better, and I think hot ends will always benefit from an insulation.