1. If the goal is looking at effects of volatility assymetry, you might want to look at Exponential GARCH (EGARCH) Models.
For reference, read Daniel Nelson's work:
Nelson, Daniel. “Conditional Heteroskedasticity in Asset Returns: A New Approach.” In Econometrica, Vol. 59, No. 2. The Econometric Society, 1991.
2. To see the length of persistence of volatility, you may easily generate a Correlogram of your time series using Eviews. The Correlogram allows you to see the structure of the autocorrelation function and the partial autocorrelation function of a given time series.
More so, the correlogram will help you identify the presence of long-range dependent serial correlation or what is known as "long memory" in a time series.
If long memory is present, you might need an ARMA specification in your mean equation. Hence, you might also want to look at ARMA- GARCH type Models.