In the episode about the neutron star, the countdown clock is incorrect. It counts down from 11yrs 1month 1day to exactly 10 yrs when it should show 10yrs 11 months 30days.
When the neutron star passes Saturn, astronomers witness it sucking up the complete gas giant within a few minutes, with its matter disappearing completely into the neutron star. Saturn's diameter is about 12,000 times larger than that of a neutron star, so this would mean all of its mass, which is 95 times the mass of Earth, would need to pass through basically a single point in space. In reality, this mass would heat up to very high temperatures due to the friction, leading to massive radiation emissions which would definitely impact Earth, causing a large-scale extinction of life already at this stage.
When the two boys watch the destruction of Earth, they are inside a hollow, cylindrical spaceship looking through a window in what is to them a wall. As visible when the camera zooms out, that wall is one of the circular ends of the cylinder. They are standing on the curved surface, toward which the (centrifugal) artificial "gravity" pulls/pushes them. They would not need to be looking through a window in their floor to see out.
The final destruction of the Earth is witnessed by two boys aboard the Ark looking through a window, but given the assumed speed of the Ark, and that it had been already traveling for some days at least, Earth certainly would not be visible to the naked eye in detail (if at all) any more.
In the opening sequence, we see meteor showers in Jeju Island, South Korea, in Marfa, Texas, and in Chile; there are also clear, starry nights. However, South Korea and Texas are 10 time zones apart. If it was dark enough for amateur astronomers in Korea, then dawn should be beginning in Texas and their star fields washed out by the dawning sun.