Theoretical models according to Black-Scholes have no skew or smile. However, because vega decreases as time to expiration decreases, I initially thought that the skew or smile would flatten. On a second thought, the value of vega bears no linear relation to volatility, vega <=> d(price)/d(vol). Therefore, I finally concluded that time has no effect on skew or smile. I.e., implied (or historical) volatility is a stochastic variable independent of time; and therefore skew or smile is not affected by time. Any comments?

Individual stocks can have unusual circumstances so I will comment on just indices such as the SPX and NDX. The shape of the implied vol curve is much more a smirk than a smile; out of the money puts have a much higher implied vol compared to out of the money calls. Over time the tails of the vol curve or "smirk" become much steeper. Simply graph the actual implied vol curve of June SPX options vs. April and you will see a significant difference.

dVega/dVol is a kurtosis/distro greek which is defined by delta > time rather than skewness. Do as SLE suggests.

That makes the axis a standard deviation of the relative ATM strike instead of a percentage. What new insight is supposed to be gained from it?

annual volatility is usually converted to the time period you're interested in. e.g., 10 day volatility (std dev) But I'm interested in hearing SLE's answer as well...

The CLT defines that the skew decreases as the reciprocal of the sqrt of duration. The skew fattens as duration increases.