Why didn't Applejack's friends and family join her in Canterlot to watch the rodeo?
Probably the family couldn't afford it. We known from her wish to make money at the Gala that the family doesn't always have enough money. This might also be part of why she was ashamed to have not won a money prize.
Okay, but what about her friends? Not only did they have enough to go to Canterlot, but they also had enough to go to Dodge as well.
Considering that it was an emergency matter, well, Applejack disappeared and they probably pulled out their savings for their friend.
It could also be that the show tickets were a lot more expensive than the train tickets.
It may have been harvest season; AJ knew that this event was coming up and the town needed her prize winnings, so arranged to have her cousins to help harvest apples while she couldn't be there.
If AJ had enough money to offer the stage coach ponies double to outrun her friends, then quadruple to counter Twilight's offer of triple, then why was she worried about returning home empty-hoofed? It seems like Miss Jubilee must pay her cherry orchard workers really well for one week of work!.
AJ wasn't acting rationally. Clearly, at that point she'd rather spend more money than she has than letting the others find out why she didn't want to come back home.
We don't know how expensive the carriage fare was. it could have just been a couple bits.
The entire chase sequence. How is a horse-drawn carriage faster than a regular, unburdened horse? Rainbow Dash can fly faster than she can run, yet she stays on the ground. A properly motivated Pinkie Pie can run faster than Rainbow Dash can fly (sort of), yet she sits in the cart. Why didn't Twilight just magic one of the wheels off AJ's cart, or make hers go faster like in "Winter Wrap Up"? Heck, Rarity could have even used her "Oh boooys!" line on the carriage ponies.
RD can't carry the cart and fly at the same time, and neither can Fluttershy, for that matter, for all that her speed is comparable to Rainbow Dash's. They couldn't even jump the train using their wings until they lost the weight of Rarity and Pinkie Pie. Team Friendship all wanted to catch up to AJ, not just one of them. Knocking the wheels off the other carriage risks Applejack and the horses pulling it, and they already tried bribing the horses to stop.
I'd say Rule of Funny. As we have seen several times, a pegasus can fly while pulling a loaded vehicle, even stopping it in mid air ("Feeling Pinkie Keen" provides two good examples).
Also, Rarity needs to be up close and personal to seduce.
The reason for Applejack's carriage is simple: as soon as she tries to run on her own, it takes Rainbow Dash only a couple seconds to pin her down. She couldn't do that while Applejack was in a vehicle pulled by four other ponies.
Applejack jumps the gun by trying to cross the tracks before the train gets there. There aren't many repercussions for her taking this great risk, never mind endangering four other pony lives into the bargain, so I've got two fridge logic moments:
Call me a Moral Guardian, but isn't this a bit of a bad thing to show on kid's TV? Crossing a train track is within their everyday experience, or at least will be when they learn to drive, and AJ is certainly the sort of character they could look at as a role model. It seems to be mitigated by the carriage ponies chewing her out for it, but still, a semi-comedic line doesn't seem like much of a punishment.
If you are old enough to learn to drive, and you still think that it's a good idea to race trains because of something you saw on a kid's show, you should not be driving. Also, where are today's teens supposed to get a horse-drawn carriage?
Presuming that last comment was an attempt at Comically Missing the Point, the fact is that people have tried crossing train tracks while the barrier is down, and some of them have been killed doing it. While it's not likely they'd be inspired to do it by watching one scene from a kid's TV show about fantasy ponies, and I'm not saying remove the scene, that action deserves a bit more than a "Lady, you're trouble," and something more like "What the hay, lady?"
Why didn't the four carriage ponies just dig in their hooves and stop? It's just as much their fault as AJ's, and they seemed to have plenty of time to slow down.
They figured they could make it. And they did.
They looked pretty nervous about it (unlike AJ, who looked gleeful at the thought of getting across in time), and chewed out AJ for making them do it. If they were that bothered, they could have stopped. Yet they didn't.
They were part of a high speed chase. Higher speed = longer stopping distance, meaning they might not have been able to stop before hitting/getting hit by the train anyway, so in this stressful situation they decided to go with what AJ said. Was it unwise? Yes, but so are most decisions made during stress.
They could've turned to the right or left, considering this isn't like a road for cars to drive on, and there were no obstacles. I agree with the stopping distance point though.
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 BSOD. 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.