No, it does not have a standard name. By analogy to the lognormal distribution it might be dubbed the "exponential-normal" distribution. But the analogy is imperfect, because you have to truncate the normal part at zero.
The integrals to compute the moments (or, equivalently, the characteristic function) do not have closed forms. You will need to compute them numerically. They are obtained by integrating
$\frac{1}{\sqrt{2 \pi } \sigma } x^k e^{x-\frac{\left(e^x-\mu \right)^2}{2 \sigma ^2}} dx$
over the entire real line. (This comes from the pdf for a Normal distribution of $y$ with mean $\mu$ and standard deviation $\sigma$ and substituting $y = exp(x)$.) $k = 1$ yields the mean; $k = 2$ gives the second moment; subtracting the square of the mean from that is the variance; and the square root of the variance is the standard deviation. Finding the width at half maximum also requires numerical methods (but convergence should be rapid).