I'm a little late to the party - finished watching Tenet after it was available to stream. I believe while the accepted answer is mostly correct, it doesn't fully address the question; what happened in Kat's perspective. Particularly No.3 above.
I'm gonna use a couple of notations here. P for Protagonist, and the '~' in front of any name to indicate an inverted person. So P is Protagonist, ~P is inverted protagonist, ~Sator is inverted Sator etc.
Let's assume you and I are outside observers who didn't participate in the entire sequence; we're obviously not inverted and witness the whole thing in 'normal' forward moving time. Let's establish a timeline and some hypothetical time stamps for events occurred for our convenience.
9:00 AM --> Sator beats up Kat and leaves her on the floor.
9:15 AM --> Neil and P successfully completes the heist and sees the car ~Sator and Kat are in, on the freeway. ~Sator has an oxygen mask on his face.
9:30 AM --> Kat and P are captured by Sator's goons, and taken into the turnstile building; Kat from the Blue side and P from the Red side. ~Sator puts on an oxygen mask to Kat's face, in awkward backward movements. ~Sator himself already has an oxygen mask on his face.
9:45 AM --> P is questioned by ~Sator from the Blue side of the turnstile building, and Kat is shot.
10:00 AM --> Ives' team bursts in, Sator (who's on the Red side) runs through the turnstile and inverts himself to be ~Sator.
So above is a list of made-up time stamps from our outside perspective of what happened. Now you asked what it looked like from Kat's perspective. This is tricky, but we can break it down. As correctly stated, Kat was never inverted at any point during the above timeline. So this means everything she experiences is the same as you experience in normal forward timeline, even when she's held hostage by ~Sator and is inside the Audi.
So based on above timeline, we know Kat was beaten up and was on the floor at 9 AM. Then while she was being held hostage by ~Sator, although ~Sator is moving backwards in time, Kat is not. So, what looks like the beginning of the car chase for ~Sator is in fact the end of the chase for Kat.
From her perspective this is what Kat sees:
- She's lying on the floor after being beaten at 9 AM.
- She's being held hostage in the car at 9:15 (here she sees ~Sator and ~Driver doing everything backward, she will see the car she's in running backwards, while she sees P in the other car doing normal movements).
- Kat will see herself captured by goons at 9:30 and then handed to a ~Sator walking backward who then proceed to take her through the airlock into the Blue side.
- Shortly after she will see ~Sator putting an oxygen mask to her face and taken into the blue side, still ~Sator walking backwards. She will see P seated in the Red side.
- At 9:45, she will see ~Sator question P, then ~Sator will 'un-shot' a bullet through her body into his gun.
- Then she'll be left bleeding on the seat, and she'll see ~Sator remove his own oxygen mask, and she'll see ~Sator disappear into the turnstile at 10:00 AM.
- She'll see a bunch of TENET team walking into the Blue side (but they are not inverted and walking in normal fashion), put her on a stretcher, and take her out of the Blue side airlock (NOT into the turnstile).
- She'll see herself being wheeled on the stretcher to the Red side, then P and Ives argue, they all decide to go in through the turnstile this time to invert themselves including Kat.
- They'll all emerge from the Blue side, say at 10:15, inverted.
Now, the real mind-bending question is how did Kat, who was on the floor in the turnstile building from 9:00 to 9:15, get into the Audi with ~Sator at 9:15, presumably miles away from where the freeport is. I think the explanation here, although not shown, is that she simply disappears from turnstile building at roughly 9:15, and appears out of thin air in the Audi alongside ~Sator. I know this is quite hard to wrap your head around because it's a person involved, but we see similar things happen to object (or even people) throughout the movie.
For instance, in the car chase itself, shortly before Neil and P sees the Audi ~Sator is in, we are shown a shot of the side mirror of the BMW that Neil is driving and we can see that it's broken. Then when the car BMW collides with the Audi, we can see the broken mirror gets repaired. So... at what point did the mirror broke before it got 'unbroken' by the collision? Well, obviously it wasn't broke from the day car was manufactures, and in fact the mirror just simply broke suddenly shortly before that. Because it has to be broke in order to get unbroken. Same way, we see that when ~P is going back to Oslo freeport with ~Neil, we see that a wound appears out of thin air on ~P's arm and he starts bleeding. So basically his intact skin suddenly got wounded without any outside contact, because it needed to be wounded in order to be 'healed' by P 'un-stabbing' himself.
Same way, once we buy into the premise that inverting entropy of an object (or person) makes them go back in time, we must also buy into the premise that objects or people can appear and disappear in order for inversion to work. This is why Kat can disappear and appear, because the events happened in inverted timeline of Sator dictated that she needed to be in the freeway at 9:15 in the car with him.
Basically, not just the characters, but we as viewers must also stop thinking in linear terms to fully understand (and appreciate) Tenet.