Is dimension reduction helpful to select features for a classification problem?
Well, let's say it depends on the distribution of your data. In approaches like PCA the approach does not care about the labels of the data in hand. This is why PCA may lead to data which are sometimes difficult to be separated or vice versa. PCA just cares about which direction leads to more variance and take that direction as a new basis. Not caring about the labels is why you cannot say it may lead to a better space for classification or not. You have to employ that and after that, investigate whether it's helpful or not. Approaches like LDA or other variants of that take care of the labels but they are linear classifiers which are not strong at least in a current feature space where you've not done any feature engineering.