Let's be charitable - Russia is a 2nd world country.
By definition, yes. That's
literally what the term "2nd world country" actually means: "Soviet Bloc." The term "3rd world country" originally referred to all of the developing countries that were not aligned with either the west or the soviet bloc during the Cold War; they were implicitly considered to be poor and under-developed mainly because neither of the major power blocs were able (or willing) to assimilate and develop them.
And no, it's not really a description of "how developed" a country is. It's actually a jingoistic term that implies that any country that hasn't enjoyed the blessings of western patronage is either Soviet Bloc (the enemy) or under-developed and not worth the effort (peasants). That hasn't been the case for over 20 years now, which is why more formal discussion on such matters now refer to "developing countries" rather than "third world" as those categories no longer overlap and arguably never did.
So no, Russia is not a "developing country" in the sense you're implying. Political intrigue aside, it's a major industrial power with a multi-trillion dollar economy. They also DO have the industrial capacity to completely leapfrop the United States in terms of space development, not to mention a lot more practical experience in that field than the U.S. currently has. There, as here, the only thing missing is the political will to do so.
China HAS the political will, and is fast-tracking the tech development to get caught up. So as a tally: we have two countries with the means but not the will, a dozen countries with the will but not the means, and one country (China) with both the means and the will. Guess which one of them is going to get the moon first?
NASA has revealed that they don't plan to build the next space station with the Ruscosmos, when the ISS is retired in the 2020s.
Assuming it IS retired in the 2020s. Roskosmos has made it clear they plan to end their participation at that time; NASA has made no such commitment and will probably be encouraged to commercialize the station by leasing huge parts of it to private operators.
NASA will work with the private sector in building the next space station.
Doubtful. It's more likely the private sector will do it themselves on their own timetable without NASA "working with" them, seeing how NASA currently has nothing useful to contribute to that effort except for technical knowledge which is already public domain. Though, again, the ISS will probably be leveraged for that purpose in some way.
Any mission to Mars will require a global effort. It's too expensive for any one nation to do.
For the moment.