About how long do astronomers think it will be until the Earth is destroyed?
Probably, about 7.5 billion years.
According to current theories (https://en.wikipedia.org/wiki/Future_of_Earth), our big blue marble will suffer multiple stages of doom. Some "highlights":
600 million years: Hot enough for you? Carbon dioxide levels drop off as a warmer earth (due to the Sun slowly becoming brighter) causes faster weathering of silicate bearing rocks (which consumes carbon dioxide). Most photosynthesis ceases leading to the demise of most earthly life.
1 billion years: Runaway ocean evaporation begins, likely leading to an end of plate tectonics as rock, deprived of water, becomes more rigid. With it goes the complex recycling mechanism needed to sustain life (if any survived the previous stage above) over the long term (https://en.wikipedia.org/wiki/Carbon_cycle).
2-3 billion years: The magnetic dynamo comes to an end. This dynamo creates the magnetic shielding that helps protect our atmosphere. When it's gone, radiation will begin to erode our atmosphere.
4 billion years: Earth's surface temperature, already hot enough to evaporate the oceans, rises more with a runaway greenhouse effect like Venus, ultimately melting the rock.
And if you think you can handle all that ...
7.5 billion years: The Sun, now a red giant, likely absorbs the Earth.