Are we required to believe that Hashem created the earth (and other planets?) in six 24-hour periods, and that the universe is literally 5772 years old?
I am not aware that we are required to believe that. We are required to believe that G-d created the world ex-nihilo, but that is ok, since there is no other logical choice anyway.
Or do you believe in evolution?
I’m not sure why you picked that as the “other choice.” I could imagine 1000 other options besides those two. For example, that Hashem created the world in 6 24 hour periods and it was 5773 years ago.