Falling off is a DQ (or zero score, depending on the class) in most divisions of Arab shows. In certain classes - mostly saddleseat style classes - a fall may or may not result in a DQ depending on the circumstances and the judge’s discretion.
Most liberty horses show in other classes. A lot are halter horses, but many of them also show in performance. The horses have no trouble differentiating between liberty time and regular riding time - that factor isn’t any different than occasionally turning out or free longing in an arena where you also ride.
I’m sure there are some horses who are alarmed by the liberty environment, but a frightened horse isn’t going to show well. These classes are usually expensive “jackpot” type classes, so there’s not much incentive to repeatedly enter a horse who doesn’t enjoy it.
The cheering is not unique to liberty classes. It’s common in halter, most saddleseat classes, and all working western disciplines. Ever watched a cutting or reining class?