What is the fate of closed Qs?
Are they eventually deleted from the site or not?
Suggestion: Shouldn't there by a penalty for the user asking the Q when his Q ends up closed?
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Closed questions never gets deleted(as far as I've seen), unless they're voted for deletion.
The question asker often gets penalty in terms of negative votes from other users, so no other penalty required. Sometimes the closed questions to attract positive vote. So it's completely on the user community whether they want to penalize the asker or not.
More info on this meta answer about deletion of closed questions.