Suppose I have trained a random forest on a very high dimensional set of features. Then I have identified the top m features that have the highest Gini index. If I'd now drop all other feature variable except the top m ones (so performing a great dimensionality reduction), can I expect to obtain a similar classification?
I don't think this question can be meaningfully answered, since this -- obviously -- depends on your data set. For example, you might have a single variable which is a perfect classifier; or you might need at least a thousand features to meaningfully classify your data. What do I know?
However, two things can be noted.
You can (and should) test how feature selection influences your models performance. Mind that to do it meaningfully, you should use a LOO-CV or KFOLD-CV testing scheme, as you will not be able to rely on the OOB (out of the bag) error estimates.
The maximum meaningful number of features will depend on the number of samples. The number of features should most likely be much smaller than the number of samples; increasing number of features beyond the number of samples will not be likely to improve the performance.