Does pre-operational exercise affect recovery from surgery? Statistical Analysis
Regarding your first question, the effect size is not invalid if the confidence interval (CI) crosses 0. The effect size is a point estimate. When the CI -a range estimate of the coverage probability- includes 0, the probability that the range of the effect size can include 0 cannot be rejected at the $1-\alpha$ level of confidence.

Regarding your second question, from the definition of Hedge's g:

$$g = \frac{\bar{x_1} - \bar{x_2}}{sd_{pooled}}$$

we can see that if $\bar{x_2}$ is larger than $\bar{x_1}$ then $g$ will be negative. So, the sign only determines which of the estimates is larger and that will depend on which estimate is considered as the reference.