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I have three related questions:
Many misunderstandings here. It would be easier to answer if you defined your terms and included some formulas. But:
1) NO, it is not correct to say that an unbiased estimator is necessarily close to the true value. It could be unbiased but with a very large variance. Its expectation equals the true value, but it could nevertheless be very far away most of the time, just that in the long run underestimation and overestimation exactly cancels. To be more concrete, say the true value is 10, and your unbiased estimator is 10+1000000000000 half the time, and 10-1000000000000 the other half. Then it is exactly unbiased, but never close.
2) I don't see the connection with 1): this is an unrelated question. But yes, if "correction" means using the denominator $n-1$ in place of $n$, then the sample variance becomes an unbiased estimator. But if $n$ is not large, it could be very far off!
3) There is a fog of misunderstandings here. You seem to think that an unbiased estimator must be normally distributed. But there is NO such connection. And, the variance is never $t$-distributed. Appropriately scaled, it might be chi squared-distributed, in the normal case. As the sample variance never can be negative, it cannot have any distribution which assigns positive probability to negative values, and that includes all normal distributions and all $t$-distributions.
