It is interesting and a bit surprising for me that some russians don't want to stand for slang phrases or "mat", acting like it don't exist and are not used.

It is indeed a very important part of learning the language, and understanding it.

Of course it also matters what you want to use the language for, but if you want to be able to discuss and act with native people, being stricly formal all the time not only makes it harder for you to understand them, it's also a bit awkward.

To conclude, I think that slang words, phrases and their meaning definately should be studied along with "normal" russian, since it's such a big part of the language.