There is no definitive answer to this question. That’s why the field of machine learning has so much empirical experimenting. I suggest to try whether it improves performance or not.

I think if you do batch Norm after dropout during training the parameters of batch norm are not correct at inference time. So I would do batch norm before dropout. On the other side it has been proven that batch norm also does some amount of regularization, so it is also fine to just use batch norm. I would choose the approach that works best for my specific use case

If you do use both in the same layer, dropout should never be applied right before batch or layer norm because the features set to 0 would affect the mean and variance calculations. As an example, it is common to use batch norm in CNNs, and then dropout after the global average pooling (before the final fc layer). Sometimes you even see dropout between conv blocks, take a look at EfficientNet by Google.

Yes, it makes sense to use dropout and layer normalization in the same model. But only if you want your model to be overfitting and perform worse than it could.