Derpy and Princess Luna share a voice actress. Derpy is Princess Luna! No wonder everyone loves her!
Derpy would be clumsy: she can't coordinate herself with those eyes of hers.
The rest of the group don't really try to reason with AJ that much until after The Reveal. Why? Remember last time they tried to reason with her when she was dead set on something? It took realizing that working herself nearly to death only got half the orchard harvested for her to finally admit she needed help. They, especially Twilight, knew Applejack is extremely stubborn when she's set on something and just trying to talk her out of staying in Dodge Junction would've been a waste of time.
Applejack actually brought her "defeat" on the Rodeo upon herself if you think about it. Consider she won A LOT of ribbons at that one rodeo, which means she had to take part in many different events. That means that Applejack has at least some experience and skill at many different events but if she spent only some of her time training for each of those different events, then of course she'd end up losing over and over again to different ponies who had spent all their time training for one particular event!
In addition, Ponyville put a ton of pressure on her, especially Mayor Mare, which clearly made her at least somewhat nervous during the part of the episode prior to her leaving and was the primary reason her loss sent her into a Heroic B.S.O.D.. It's completely possible all the pressure made her choke at the competition.
And lastly, a main reason why she could have lost one or more of the competitions is seen at the very beginning already. When she jumps the extra-high hurdle, her hind hooves strafe it. What if she did the same thing at the actual competition? Even though it was only a minor slip, it could have still either been a plainly illegal move (coming into contact with the hurdle), or may even have resulted in a much bigger mess-up (losing her balance, flying off-course, tripping, etc). After all, Applejack apparently didn't bother to eradicate this one misstep in her otherwise completely flawless training.
The board with five blue ribbons just after Applejack is announced as a ten-time champion is pretty much explained in-universe - AJ's habit is obviously in taking on all the events each time, and the five ribbons were all from the same outing. And if it's true about other ponies only competing in one or two events, five blue ribbons is a lot. Which sheds some light on the way things turned out - most of the other competitors would probably marvel at her ability to get one medal for every event at all, but only the hoity-toity winners would make sure to point out that none of them are blue, explaining all the added pressure on her.