The auto mode does several things. It adjusts the A/C unit's output and the fan speed as needed to reach the newly set temperature. And it continues to adjust these things as exterior conditions (sunshine, external temperature, etc.) change the loads on the system.
Usually, when you adjust the temperature while in auto mode, it's just because you want to be a little warner or a little cooler, but you don't want the car to quit adjusting the fan and A/C output to keep that new temperature you have chosen. Right now, just adjusting the temperature up or down a degree or two means that, unless you go back into the lower screen menu to re-engage auto mode, you're also taking on the need to adjust other things manually as cabin temperature drifts up or down, which is probably not what you wanted to do. (You have the auto mode "off" switch to do that.)
In our Honda, for instance, we keep the temperature set at 70 in auto mode. Then, if we're dressed unusually heavily or lightly or are a bit hot or cold from staying outside for a while, we just tap a toggle switch to raise or lower the temperature accordingly. The auto system stays engaged and continues to make the necessary fan and A/C output adjustments to hold that new temperature.
Yes, it does. I found this out only today. We went to the botanical gardens and did a lot of walking. The car was parked in direct sunshine, and the A/C did not come on strong when we got back in. We were both sweating a bit and toggled both switches to higher temperatures. Nothing happened. I thought it might be another one of the excruciatingly slow responses of the Lucid, but as we continued to drive, we just kept getting hotter and hotter, with no air coming out of any of the vents. We toggled both fans up to "10", and still no airflow. It was 70 degrees outside, and we usually keep the auto mode set at 70, so I thought that the A/C's programming might be having trouble figuring out what to do with a hot interior and a cooler outside. Finally, we had to let the windows down to make the car bearable.
As the interior cooled off and we raised the windows, I thought things might work again. But still no air out of the vents, and the car started heating up again. Finally, I opened up the A/C menu on the Pilot Screen and found that both sides had been taken out of auto mode -- and that the whole A/C system had been turned off. I then remembered that, just before arriving at the botanical gardens, my passenger had adjusted the temperature on his side with the toggle switch. I turned auto mode back on, and the system fired right up and began to cool the car. With everything now running properly, I decided to toggle the temperature on my side to see what happened. When I went back into the A/C menu, sure enough, the toggle had taken both sides out of auto mode.