They say the planet orbits a dwarf star that will live longer than our sun. Would that star have once been hotter and is now a dwarf star at the latter end of its life cycle? If so, then wouldn't any chance of water or atmosphere be much lower?
Gliese 581 is an "M Type" star, more colloquially called a "red dwarf". This is not a star at the end of its life, but just a star that started out small to begin with (this one is about 0.3 solar masses). The bigger a star is, the hotter it is in the middle, due to the increased pressure. the hotter it is, the faster it burns through nuclear fuel. So, big stars don't last long, small stars last longer.
What about other radiation? Being that much closer to the star, would there be higher radiation or is it all proportional to size and visible light intensity?
How much "radiation" you get on the surface of the planet depends on many things, perhaps most importantly on the nature of any magnetic field the planet has, and the planet's atmosphere.
Yes, the planet is closer to the star, but it's a cool, placid star.
The article states that the planet does not rotate much, so one side is almost always, the other dark. How do rotation and orbit correlate? I'm just thinking about our moon, with a rotation and orbit that are both about 28 days, so we always observe the same face from Earth.
Yes. It's called "
tidal locking", and the situation for this planet is just like for our Moon - one face always toward the object it is orbiting.
It may be best to think of this not relative to the Sun or Earth, but relative to the distant stars (which don't move appreciably over this kind of timescale). Set up a telescope on the Moon, and have it point directly overhead. As the lunar day goes on, the view the telescope gets sweeps across those distant stars. 28 days later*, you're pointing back at the same chunk of distant sky you started pointing at.
*I'm leaving out some details, but this works for first approximation