what are the mean and variance of the sum of two folded normal distribution? 1) Does sum of two independent folded normal distribution always yield a folded normal distribution? (if independent random variables X and Y follow normal distributions, then does $|X| +|Y|$ equal to some folded normal distribution?) 2) If so, then given the mean and variances of those two independent folded normal distribution, what are the mean and variance of the sum of those distributions? (Suppose $|X|$ ~ $FN(\mu_{x},\sigma_{x})$, $|Y|$ ~ $FN(\mu_{y},\sigma_{y})$, then what are the mean and variance of $|X| + |Y|$ in terms of $\mu_{x},\mu_{y},\sigma_{y},\sigma_{x} $ if possible?) I looked at other sources but it seems folded normal distribution is something not talked about a lot.

To answer (2), we still have linearity of expected value and variance, i.e. $$ \mathbb{E}[|X| + |Y|] = \mathbb{E}[|X|] + \mathbb{E}[|Y|] $$ and same for variance. You should be able to find the mean and variance of the folded normal from the parameters of the original normal using the definition, e.g. $$ \mathbb{E}[|X|] = \int_{-\infty}^\infty |x| \phi(x; \mu, \sigma) dx... $$