1) Is normalization necessary for Random Forests?
2) Should all the features be normalized or only numerical ones?
3) Does it matter whether I normalize before or after splitting into train and test data?
4) Do I need to pre-process features of the future object that will be classified as well? (after accepting the model, not during the testing)
1) No! Feature normalization isn't necessary for any tree-based classifier.
2) Generally speaking, normalization should be done on all features not just numerical ones.
3) In practice it doesn't make much difference. However, the correct practice is to identify the min and max values of each feature from the training set and then normalize the features of both sets according to those values.
4) Yes, afterwards any sample that needs to be classified should be processed with exactly the same way as you did during training.