Yes taking them out of the turner is what probably killed them. You never want to change the position of the egg. Just turn off the turner so they don't get caught in it, OR you could get an egg carton and put them in it and just put them in the bottom of the bator. But you want them upright to hatch since the turner holds them upright.
The temp is the same in mine. The temp is set at 102 and reads 99 inside, so that's what I keep it at. When you open the door and then close it, the bator works to bring the temp back up, so that's why the outside thermometer reads 102-103 temporarily, don't worry about that its just because its heating up to compensate for lost heat. Just keep an eye on the thermometer inside, if it ever reads high then you have an issue.
What you are describing as far as temps go is exactly what mine does
I wouldn't worry about the momentary drop in humidity. But if it raised to 84 then dropped back to 65 then that's fine. As long as the humidity that's holding isn't above 70-75 you should be okay.
But yes I would say laying the eggs down is what killed them so close to hatch, not the momentary spike in humidity.