No.
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Let sigma = standard deviation. Standard error (of the sample mean) = sigma / square root of (n), where n is the sample size. Since you are dividing the standard deviation by a positive number greater than 1, the standard error is always smaller than the standard deviation.
No. The average of the deviations, or mean deviation, will always be zero. The standard deviation is the average squared deviation which is usually non-zero.
Yes; the standard deviation is the square root of the mean, so it will always be larger.
Mean 0, standard deviation 1.
Mean = 0 Standard Deviation = 1