Should we also include negative instance in cross-validation process of one-class classifiers?
The evaluation design depends on the goal, i.e. how the classifier is intended to be used in production. The goal of a one-class classifier is to find instances of this class among other instances, so it has to be evaluated with data which includes negative instances. The choice of the proportion and type of negative instances should be driven by the final goal.
Note that in terms of evaluation and cross-validation the setting is the same as regular supervised classification, the difference is only that negative instances are not used during training.