Crisiss I figured the 3 way war was unlikely but put it for two reasons. 1). It is all but assured that we would win the War but even victories can wind up being too costly. Take Britain in World War 2 for example. To win they had to purchase ships and supplies from America but had to give up pretty much all of their Naval Bases, including Iceland, allowed them to rule the oceans and the world for so long. Also they wound up being hugely in debt and not able to sustain their Empire in Asia and Africa leading to the loss of that. Something like this could happen to America fighting 1 billion Chinese and a crap load of Rusky tanks. We would be tied up in Europe trying to save our allies there and need an Ally in the Pacific so get India to help out. To get them to join us we allow them to take Sri Lanka, Taiwan, Indonesia, and the Philipines(?) giving them an Empire to build upon. After the war we are exhausted finically and wind up giving up our military bases in foreign countries and gradually turning into a second tier power. 2). I could not figure out how to get rid of options so I put random ones in.
You said this was a 3 way war though. Russia vs China vs America. None of Europe other than western Russia. And no, we would really only have to pay operating costs for our navy and aircraft to keep them at bay. So unless they plan on swimming across the Pacific or the Bering Strait (or going over the North Pole for whatever reason) we would be well prepared. But if we did launch a land offensive, we would get screwed financially. I don't think Europe would want to join anyways as nobody wants to piss off Russia. We all know what happens when you piss off Russia. China would have a hard time supplying and moving 1 billion troops and billions of tons of equipment across the Himalayas and Middle East, not including food, fuel, and ammunition. In a sense, nobody would win because of the economics. America has the standard victory by destroying the opposition down though.