FWIW, "Miri" shows our heroes doing the "standard" approach to the planet that proves to be a Second Earth. That is, their ship is seen from behind as she approaches the planet, just like in the opening credits, and then the ship curves to the right, seemingly to orbit the planet counterclockwise, or counterrotationwise, or whatever you want to call it when you go towards the sunrise. Kirk then calls for "fixed orbit". And the very next thing we see on the viewscreen is the continents floating past, from left to right, as the ship travels from east to west!
Certainly suggests that starships don't respect freefall orbits much.
Any descent into the atmosphere can be avoided by a powered ascent to a circular orbit which should remain stable for days, weeks, months, or longer, depending on the altitude.
And even a badly hurt starship whose crew is unconscious can manage this easily enough, as in "Tomorrow is Yesterday". Yet a starship that suffers a loss of power due to human sabotage or alien influence cannot. Which is probably because the malevolents specifically hurt the ship in a manner that makes the use of propulsion of any sort impossible. In which case any powered orbit quickly becomes fatal enough.
If the orbit is decaying rapidly due to atmospheric drag, then the ship would be heating up and the hull ablating.
The hull tolerates atmospheric flight easily enough, it seems. And speeds would be fairly low if the ship merely fell straight down towards the planet when her hovering power failed - the first hour could well be spent just drifting down through vacuum, from the apparent height of several thousand kilometers.
Well, if it's an extreme range sensor array then you'd never point it at anything less than many AU's distant.
True enough. It wouldn't do anything plot-relevant, then, but might conduct scientific research despite
the ship being engaged in the plot of the week.
It sounds a bit unlikely for this seemingly "lean and mean" ship to carry such gear, but by no means impossible. Perhaps this is the technology that was used for taking the star photographs in "Corbomite Maneuver"? Although plotwise, we have never heard of a sensor that would be pointed by pointing the ship; when the planet below or the ship ahead is scanned, this takes place at fairly odd orientations (typically planet to lower port, ship of interest to upper starboard bows), and the viewscreen shows angles unrelated to the orientation of the hero ship, too.