astrophysics - Do the stars imaged by a telescope even exist at present?

I know that we now have telescopes which can capture the images of the stars and galaxies millions of light-years away from us.

Does the telescope capture the past image of the star, i.e. the light which it emitted centuries ago?

What guarantee is there that the star is still alive?

What basis do organizations like NASA plan missions for evading such stars?

Answer

For a star of a given mass, we can calculate, based on theoretical models, how long it has to live. For example, the Sun is currently 5 Gyr old and will live another 5 Gyr. So if you observe the Sun from, say, Andromeda, the light would be about 2 million years old and you could therefore conclude that the Sun is still alive even though you're observing "old" light. If you flip it around, if we see a Sun-like star in Andromeda, we could safely say it's still alive.

Broadly, while fusing hydrogen in their cores, more massive (and therefore brighter, hotter) stars live shorter lives. On the one hand, it means that we observe some small stars whose lives will be longer than the current age of the Universe. On the other hand, we could theoretically observe stars whose lifetimes are shorter than the distance to them, in light-years. So, a star 80 times as massive as the Sun might have a lifetime of a few million years, so if we currently observe it in Andromeda near the end of its life, that star is probably gone now. Off the top of my head, I'd say most of the stars we presently observe still exist as we see them now. We just can't resolve stars that far away.

For what it's worth, there are some distant phenomena that we see that happen much faster than the light takes to reach us. For example, the 2011 Nobel Prize went to the leaders of two teams that observe Type Ia supernova at redshifts up to about $z=1$. These are events that lasted less than a year that happened nearly 8 billion years ago. Who knows what they look like now!