The Anthropocene Is a Joke
theatlantic.com
theatlantic.com
It's not clear how the Earth will end up, since human population reduction looks unlikely barring catastrophe - intentional or otherwise.
However, I think humans will very quickly (< 100 years) spread throughout the solar system, and eventually into the Oort cloud. At some point, sooner or later, humans will also become an interstellar species.
Human artifacts may or may not always be visible on the Earth, but they'll be easily visible on the Moon until it gets eaten by the Sun.
On that note, it's worth mentioning that life on Earth is roughly 4/5 of the way along its timeline. Earth will become unbearably hot in well under a billion years, regardless of human contribution, due to increasing solar luminosity...
So we can look forward to a world full of dry, barren farms going into a hot spell. Heat of summers is getting lethal in places; expect to see more of that. I'd say human population reduction looks extremely likely, due to the ongoing climate catastrophe.
This may be our filter. Maybe we'll live on. Maybe a science-literate subset of us will live on. But devastation seems much more likely than sustainable colonization of space.
Those aren't mutually exclusive outcomes...
At any rate, I think global warming is overrated as a threat. Market forces will soon greatly reduce CO2 production - it mostly depends on how quickly China and India can be weaned off of coal.
The other big issue is having a worldwide massive nuclear buildout so we have reliable, zero-carbon energy to back up unreliable "renewables". Eventually, it might be fusion, which will also produce plenty of radioactive waste. (None of the current designs are aneutronic.)
Regardless, permanent human colonization of the Moon is relatively easy, and the technological gains there should make asteroid mining feasible. At that point, there'll be so much money to be made that there'll be an explosion into space.
To that extent, human CO2 emissions may be beneficial in the longer run!
But imagine if civilization had ended 500 years ago, or even a thousand. There would be absolutely nothing to show for it in a couple hundred million years time.
I think some stone artifacts (cities, pyramids, roads) would still be identifiable in some cases. Excavation would be needed, of course.
Similarly, some of our concrete, glass and non-ferrous metal construction should survive a long, long time.