How to use random forest model to new data?
Yes, you can do that. However, how accurate your new labels are depends on the ability of your model to generalize to new data and the similarity of the new data to your training data. Therefore, it is something you need to test for in the first place by using a separate test dataset to assess model performance. In contrast to that, a validation dataset is used for model selection and hyperparameter tuning (e.g. max depth of your trees).  

An in-depth coverage of the topic you can find in chapter 7 of "The Elements of statistical Learning" by Hastie et al.  

With regards to your second question: If your training data does not contain the variables prop_C and prop_D then your model cannot make use of them. So either include these in the training data or ignore them in the new data.