During rush hour, it can be very stressful to get off if you get pushed to the back of the train, but there is an unspoken procedure. If you need to get off soon, you simply tap the person in front of you on the shoulder and ask them if they are getting off. Often, they won't be so they will turn sideways and allow you to slide past them. You continue this until you are either at the front of the crowd or close enough that you are with the group getting off at the next stop.
After a few weeks, I came to love the Metro. It is incredibly cheap, less than 30 U.S. cents a trip. It is also very clean and mostly safe because there are usually a lot of people around. The only fear that I had was getting pick-pocketed, but thankfully, that never happened. I traveled on the Metro countless times alone and with friends, and I never felt any real danger!
Yes, public transportation is very connected to Russian culture. Russians from Moscow and St. Petersburg are incredibly proud of their transportation systems and how well they run. During the Soviet period, the government focused heavily on improving the infrastructure of city areas but didn't look outside of major population hubs. As a result, you have a stark contrast between the clean and well-organized Metro of Moscow and the single-lane dirt roads that connect the majority of rural Russia. I hope that the Russian government will invest more heavily in its infrastructure in the future, however, as Russia becomes more involved abroad, it seems unlikely. It would make Moscow and St. Petersburg more accessible, even though major population centers are still connected by the Trans-Siberian railroad.