You don't have to shift down. Just let the car do that part. Shifting down a couple of gears to pass quickly is a nice option though.
The thing I like about using the paddles is a feeling of engagement, coming from years of half my vehicles having manual transmissions. My first car, a Bug, was of course a stick on the floor. Loved that about it. If the Stinger had a stick option I would have insisted on it.
"Accurately" shifting is any time before you pass 5000 RPM, because if you wait that long the car will upshift for you. So, if you engage the turbos be ready to pull the paddle or the car will call you an idiot by upshifting on its own.
The other thing I like about using the paddles to hold onto a gear is that the trans is shifting less often, which has to help prolong the life of the trans, but this is only a thing if you plan on hanging onto the car, which I do.
Low transmission fluid will explain everything that you've noticed. I know that you feel bad already. I doubt that you've damaged anything.
As far as temperature goes, the auxiliary gauge is for engine oil temperature, not trans fluid temperature. Do you have an aftermarket transmission temperature sensor?
Anyway, as far as the oil warming up goes, mine definitely takes longer than ten minutes in colder temperatures to get up to c. 200F. This is for the GT1. The Premium does not have a "Gauges" screen, and only has the coolant temperature gauge. Interestingly, this sole gauge takes at least twice as long to move as the same gauge in the GT1. I don't know this, but assume that the difference is tied to the lack of an oil temperature gauge, so that the only gauge the Premium has functions as an indicator of both oil and coolant coming up to operating temp. But it could just be a difference is individual gauges and have nothing whatsoever to do with oil.