The likelihood of a full out war between the US and Russia is very unlikely. The closest border the two nations share is over the Bering Strait but that's not an important location for either nation. If Russia was still insistent on fighting the US it would have to fight in Europe against US allies (which is dumb). Russia has nothing to gain from that since they're such a weak nation militarily and would get destroyed by NATO. The only logistically realistic war would be a nuclear one and since no one is bent on destroying the human race, I doubt we will see that.
Since the likelihood of a Russian US war is slim to none, let's look at how China operates with its "soft-power". The Chinese strategy for acheining hegemony in the East is to leverage its economic power on other states and make them dependent/loyal. Although not in Asia, the Chinese has applied this strategy quite notably in Africa (for instance they are building the African Union Headquarters) and its given them access to good business deals. If they were to "take every country in Asia one by one", first they would lose a lot of the trade and business deals with other nations and second, (since Russian and the US won't go to war) this would just be seen as expansionism. The Chinese want to spread their economic power across the world—their soft power.
Also, there are plenty of treaty organizations in Asia to prevent any Chinese expansion (US alliances with Korea, with Japan, SEATO, etc).