Should you perform feature scaling/mean normalization when predicting with a model? Should you also perform feature scaling/mean normalization when predicting with a model, that was trained and tested on with feature scaling/mean normalization?
Yes, testing data should follow the same preprocessing as the training data. Otherwise, testing data will have nothing comparable with what the algorithm learned, leading to (very) bad performances.

note: In Sklearn, the Pipeline class helps you to respect the fundamentals of ML modeling like data leakage and applying the same transformations to train and test set.