Why they look red
Their surfaces are cooler than the Sun, commonly around 3,300-4,000 K. Cooler surface temperature shifts much of the visible light toward red and orange.
They are still extremely luminous because their radii are enormous.
Why they are unstable
Red supergiants have extended atmospheres, large convective cells, and strong mass loss. Their brightness and apparent shape can change with time.
This instability is one reason radius estimates for stars like Betelgeuse and UY Scuti are given as ranges.
What happens next
Many red supergiants are expected to end as core-collapse supernovae. Before that, they can shed gas and dust that enriches nearby space.
The remnant depends on core mass and collapse physics.
What radiation do red supergiants emit?
Their cool surfaces emit a large share of their thermal radiation at red and infrared wavelengths. Dust in their extended atmospheres can add even more infrared emission.
Cool does not mean dim. Their enormous surface area allows red supergiants to produce tens of thousands of times the Sun's total luminosity.
How fast do red supergiants rotate?
There is no universal rotation rate. As a massive star expands, conservation of angular momentum tends to slow its surface rotation.
Large convective cells, broad spectral lines, pulsation, and mass loss make rotation difficult to isolate and measure precisely.