How does the idea that the universe is dying (trumpeted as some big new revelation the past few days on several new sites I keep seeing pop up on google news) square with the idea that the universe is ever expanding?
Both are true.
The universe is "dying" in the sense that stars eventually run out of hydrogen, and there aren't infinite amounts of hydrogen in galaxies to replace old stars. This will take a very long time, but eventually all stars will burn out and new stars will stop forming. This is what the study recently in the news is saying.
The universe is ever expanding in the sense that distances between galaxies which are not bound together gravitationally is expanding all the time. If you look at very distant galaxies, they are all receding into the distance.
Combine these two facts and the result is that the universe will become more and more "rarefied" (distances between galaxies will keep expanding), and also all existing stars in galaxies will burn out over trillions of years.
There are various theories prior to big bang. Some say after the big bang the universe is ever expanding. Some say the universe is contracting and once will explode into supernova. And there is a third theory that universe is like a heart. For a moment it expands and then contracts and this is a continuous process. But research now says that the theory of expansion is correct as the heavenly bodies move to the red area of the spectrograph. That is the distance between us and them is continuously increasing.