Positive or negative impact of features in prediction with Random Forest
With decision trees you cannot directly get the positive or negative effects of each variable as you would with say a linear regression through the coefficients. Its just not the way decision trees work. As you point out, the training process involves finding optimal features and splits at each node by looking at the gini index or the mutual information with the target variable. But no parameters are learnt during the process which we could use for such analysis.

A common tool that is used for this purpose is SHAP. In fact, there is a specific explainer for decision trees based models which is the TreeExplainer. With SHAP you can get both the contribution of the features in pushing towards one or another value of the label, and also an overall view of the contribution of all features.