You should use proper scoring-rules for probabilistic predictions. Take a look at the tag wiki for more information.
Log loss is a proper scoring rule. I personally would use this. The mean squared error between probabilistic predictions and the actual is the Brier score, which is also a proper scoring rule, so this would also be a possibility. As a matter of fact, both scoring rules are actually strictly proper. Our thread Why is LogLoss preferred over other proper scoring rules? compares the two.
Taking the root of the MSE will not have an impact on its properness. With normalization, it depends on how you want to normalize.
The AUC, as a scoring rule, is problematic.
Accuracy is an improper scoring rule. More precisely, it's not a scoring rule at all. Don't use it. See also Why is accuracy not the best measure for assessing classification models?
My personal favorite would be the log loss, with the Brier score as a close second, for reasons given in the thread linked above.