Betelgeuse's distance from Earth according to various well-respected astronomical agencies is between 310 and 640 light years. Taking an average of the ten measurements that I've dug out the average is 480 light years.
As I'm writing this from England and the distance to Betelgeuse is taken from the Equator, it's average distance is actually 480 light years PLUS 3,521miles!
It is a cool star.
Betelgeuse is one of the larger stars, while Barnard's star is just a little red dwarf star. So, yes- Betelgeuse is far larger than Barnard's.
The light coming from it takes hundreds of years to reach us, so what you see when you look at it is light that left it hundreds of years ago. So if it did explode and die, then it would take hundreds of years for us to see it happen.
When we look at stars, we are seeing the light that they are emitting. That light could have left them many years ago. The light from Betelgeuse takes over 640 years to reach us. So if it exploded today, we would not see that explosion for over 640 years. So it is possible that it is already gone, but the light that left before it went is still coming towards us.
No. It is in the Milky Way galaxy, like all the stars that we can see are.
It's a fairly cool star
When talking about stars, "cool" is a relative term. Betelgeuse has a variable surface temperature of 5,200 to 6,100 degrees Fahrenheit. Such a temperature may seem extremely hot to us and is easily hot enough to emit bright light but it is fairly cool as stars go. The comparatively low temperature of Betelgeuse means that it gives off less light per square meter than a hotter star would, but it makes up for it with its massive size, more than 900 times the diameter of the sun. As massive size means a large surface are and thus a huge output of light.
The density of Betelgeuse is extremely low: 1,119.10 ex.-8 from the density of the sun.
No. While Betelgeuse is larger and brighter than the sun, it actually has a lower surface temperature.
Yes, the Sun is much smaller than Betelgeuse.