My thoughts on some issues-
Disappearing on radar, my understanding is that with all their gear on, to target the Russian ship among other things, the US ship creates a blip visible to the Russians in whatever scale they are looking at. You know how you can zoom in on a minimap in a game to make locations larger? Like that. So by shutting down everything they are emitting, they reduce the size of the blip they make. Meanwhile, the Russians are still looking for a certain scale of blip, which the aluminum provides. It may not work that way in real life but it works for me as a story device. The captain even says something like "They'll see us like any other ship here", implying that they will not be invisible to radar, simply a different size target from what the Russians expect.
About not blowing up the Russians, I think that's due to the mission priority. Creating this vaccine or cure, that's it, that's the whole ballgame for the US ship. Engaging in an unnecessary ship battle, even if you have an advantage, does not serve that purpose. You're risking too much attacking them.
One thing I was thinking about, the Russian commander has a good point. Human civilization, as far as I can tell, is effectively destroyed. Let's say you do create a vaccine, now what? If you're on that US ship then you know the US government and military no longer exist, that's why you can't get anyone to respond to your ship. Who are you serving if not yourself?