First off, I'm not sure who your audience is. At some point the text seems directed towards little girls, at other times towards adults (sometimes specifically towards bronies). The styles don't really fit well together, IMO. It kinda gives the feel that the document pretends to be aimed at the little girl, but really is written for bronies.

I'm much more interested in this being a kid's game than a game for bronies - bronies already have Ponyfinder and too many other fanprojects to count - so I would very much appreciate any specific examples you can give me so I can fix it.  Part of the problem is, like it or not, I'm an adult fan of the show that got immersed in the fandom - probably have habits that are hard to break.

Rules-wise:
Make Someone’s Day: The 7-9 result seems weird and a bit bland in the context of the move. Weird because "she’ll ask you what you want her to do." when the move is not "Asking Her For A Favor", and bland because gaining experience seems very neutral, when you are cheering someone up. I'm thinking advantage  or bond would be a better pick.

Advantage or bond would be a better pick, you're right.  You're also not the only person to mention that.

Part of the problem here is that I started out by adapting the Maturity moves from Monsterhearts as basic moves, in this case Make Someone Feel Beautiful - and Monsterhearts is a much more adversarial game, even when the characters grow up.  Call Out A Cruelty is likewise a renamed Call Bullshit, and Valor's In Deepest Confidence is a reskinned Share Pain.

Bonds: I'm thinking you might want to flip the order in which you tell about the function of bonds. So first explain how you Aid a Friend, and then that you get xp at three bonds. So rather than being boiled down to, "bonds turn into xp, and you can also use it to Aid a Friend" it becomes "Bonds are for helping friends, and if you get too many, one of them becomes xp."

This is much more in line with the function of Bonds, yes.  They're primarily for the fantastically good 1d bonus to aid.

Warm. Kind, patient, beautiful, charming, empathetic.  Used to Persuade or Manipulate and to Make Someone Feel Beautiful.

Cool. Self-assured, willful, graceful, careful, steady, courageous.  Used to Keep Cool Under Pressure and to Call Out Bad Behavior.

Tough. Strong, valiant, hardy, powerful, healthy.  Used to Fight For Something and to Stand In Defense.

Sharp. Clever, intelligent, logical, observant, curious.  Used to Get The Measure Of Someone and to Stop And Think About The Situation.

Magical. Fey, intuitive, musical, insightful, just plain humming with energy.  Used to Open Your Heart to the Song of Magic.


When you persuade or manipulate a NPC with either a reasonable alternative or by exerting leverage, roll+Warm.
On a 10+, they do as you suggest, with only a verbal reassurance needed.
On a 7-9, they need some concrete reassurance of your intentions, right now, before they'll take your advice.
On a 6-, they don't take your advice. They might even get suspicious, asking themselves why you'd ask that and drawing their own, worst, conclusions.
 
It has to be a reasonable alternative if you persuade.  That is, you can't suggest that someone with a family to feed just up and quit a tyrant of a boss; what would they do from there?  If you've got a job offer for them, well then, that becomes a reasonable alternative!

Likewise, if you manipulate, you need to have leverage - something hanging over the other party's head.  A fire-breathing dragon is going to laugh at you if you just tell them to leave, now; you're a pretty little horse, it could flash-fry you and gobble you up as a snack absent some leverage.  You need to get its attention with a carrot or stick, first.  Well, dragon; a huge gemstone or a scale-slicing sword would work better.
 
When you make someone feel beautiful, roll +Warm.
On a +10, choose any two:
·         They carry +1 forward, or gain Advantage if they're an NPC.
·         They can remove a condition.
·         They mark experience.
On a 7-9, you may offer them an experience point, or a Bond if they are an NPC, if they do what you want.
 
Interpret 'beautiful' as broadly as you like, but however you define it you have to be sincere.  Otherwise, you're just persuading or manipulating, as above.
 
When you stay cool under pressure, roll +Cool.
On a +10, you don't even break a sweat.  You do it and you do it well.  Choose one: Take +1 forward this scene, remove a Condition, or put a condition on the scene or anyone opposing you.
On a 7-9, choose: either you barely keep your cool, or you pick one of the above options and you also get a condition of the MC's choice.
On a 6-, you can't keep your cool.  You melt under the pressure.
 
Stay cool is basically a catch-all for anything requiring focus, time, or courage to deal with.  The MC will say what the pressure is, in those words: “Keep cool under pressure of opening the door before the Ice Trolls reach you,” or “keep cool under the pressure of this nitwit insulting you.”

Usually the conditions you pick up will be Angry, Hungry, Terrified, or Exhausted.  Sometimes something else is more appropriate.
 
When you call out bad behavior, roll+Cool.
On a 10+, choose two.  On a 7-9, choose one.
·         They act at -2, at Disadvantage, or fail outright at the villainy they tried to wreak.
·         You lay a condition on them.
·         They must keep cool to retaliate.
·         You can offer them an experience point to do what you want.
 
A call-out doesn't need to be made in a spirit of outrage, but it needs to directly address what you think was wrong behavior: “Thunderclap, you apologize to him right now!” is calling out, but so is “I’m just a bit disappointed that you didn’t tell me this sooner.”
 
When you fight over something, roll +Tough.
On a 10+, choose to deal harm or lay a condition and also choose two:
·         They need to Keep Cool to retaliate this scene,
·         However much harm you'd deal, deal one more harm;
·         You can remove a condition from a friend,
·         You can take hold of what you're fighting over.
On a 7-9, you deal harm but also choose one:
·         They lay a condition on you;
·         They can deal harm to you.
 
This is a physical fight, slashing hooves or spears or swords, ugly bruises and cuts.  It's never pretty when one of the pretty little horses needs to fight.  On the other hoof, what you're fighting over can be a thing, or a place, or a person, or even something abstract like 'the initiative.'
 
When you stand in defense over a friend, thing or place, roll +Tough.
On a 10+, get 3 hold.  On a 7-9, get 1 hold.  Spend your hold, 1 for 1, to do any of the following:
- Redirect the attack from its target to you.
- Halve the damage of the attack.
- Open up the attacker to a counter-attack from a friend, who takes +1 Forward to attack them.
- Lay a condition on the attacker.
 
Again, this needs to defend against actual violence.  Someone can stand in defense over you if you fight over something and an opponent tries to deal harm to you.
 
When you get the measure of someone, roll +Sharp.
On a 10+, get 3 hold over that being.  On a 7-9, get 1 hold on that being.  On a -6, get one hold on them, but they get one hold on you.  Spend hold, 1 for 1, to ask any one of these questions in the course of your conversation:
·         Are you telling me the truth?
·         How are you really feeling?
·         What do you want to do?
·         What do you wish I’d do?
·         How could I get you to do (something)?
·         What do you consider beautiful and good?
 
When you stop and think about the situation, roll +Sharp.
On a 10+, ask any three.  On a 7-9, ask any one.  On a -6, the MC gets to ask you any of these questions and you must answer honestly, and may make as hard and direct a move as they want.
·         What happened here recently?
·         What should I be on the look-out for?
·         What here is beautiful and good?
·         What here is not as it seems to be?
·         What is my best route of escape?
·         What’s about to happen?
 
These two moves are very similar.  To get the measure of a being, you must engage it in conversation, and you must ask your questions as they come up; but you don’t have to ask directly.  You can always go, like, “Oh come now, we’re all unicorns here (I’m trying to get her to say how she REALLY feels)” and that’s well and good.  When you stop and think about a situation, likewise, you're analyzing your surroundings and the events that led up to this moment.
 
MC, you must answer honestly.  If this means the answer is ‘Nothing here is beautiful and good, this place is screamingly wrong, get out of here now” then that’s the truth.  Likewise, “Everything is as it seems.  What you see is what you get” is also fine.
 
But honestly?  This is All The Pretty Little Horses.  There’s probably something beautiful and good there, even if it’s bent or broken or wounded.
 
When you open your heart to the song of magic, roll +Magical.
On a 10+, the magic of the world sings to you.  You get a clear sign, a clear answer to your problems, in the call and response of the chorus and the rhythm of the world’s heartbeat.
On a 7-9, it's not as clear.  The magic of the world sings, but fainter.  It leaves you with questions - the right questions, mind - but no answers.  Alternatively, you could get the answers you seek, but gain the condition Exhausted from dancing to the beat of the song of magic.
On a 6-, the discord of the world rushes in while your heart is open and vulnerable, and the song it sings is horrifying indeed.
 
If the answer can be expressed in music, in both song and dance, it should be.  That said, any sign or portent works, so long as it's bright and beautiful.  And yes, the musical metaphor is an excuse for the musical numbers so beloved of my source material.