I don't think I've ever seen a randomized order of competition for every event. I have seen the following: alphabetical by last name, alphabetical but divided into age groups, somewhat random and either divided by age or not, coaches select one order that carries through entire meet, and coaches select order for each event. I suspect but cannot confirm that the random ones are either order in which our office manager entered the gymnasts or USAG member number. Within each of these systems except for "coaches set order for each event," some meets have the rotation start in the same place and some have a different child kick off the rotation for each event so that the same kids are not first and last on every event at the meet. At boys' states, the coaches traditionally have the guys compete in reverse order of their scores at the previous meet, with some tweaks on events like pbars that require re-setting the equipment. (Yes, when I get bored watching warmups or waiting for a rotation to end, I busy my mind by figuring out the logic behind the day's competition order.)
In all of this, I haven't seen any significant evidence of a later in rotation-higher score correlation that is not covariant with the expected skill level of the gymnast. Mind you, I have not actually gathered the data and performed the statistical test, but that's my general observation.