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Matt
I am not a professional astronomer, just an enthusiast, but I will explain it as I understand it. Red stars come in several varieties. The most common are Red Dwarfs, which comprise the majority of stars in our galaxy. These are low mass stars burning hydrogen and fairly low temperature - hence the red colour.Because they are burning hydrogen they are main sequence stars. Because they are low mass, they burn very slowly and have potential lifetimes of 100s of billions of years even into the trillions of years for smaller ones.
These are very far from dying or dead stars the Universe is only 14 billion years old so none of them have come close to exhausting their fuel
Other red stars are the red giants. These are stars of roughly similar mass to our sun or slightly larger, that have exhausted their supply of hydrogen and started burning helium. As this requires higher core temperature and the energy output is greater, the outer atmosphere expands massively. The sun would expand to beyond the orbit of the earth!
Then you have carbon stars. Classical carbon stars are red giants with large amounts of carbon in their atmosphere.
Most stars that you will see in the sky are not significant redshifted by their motion away from us. As they are in our own galaxy the speeds would be too small to be easily dedected except with pretty sophisticated gear. Redshifts become significant when observing other galaxies where the distances and speeds are much greater.
Malcolm
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