Joe Exotic may have committed a crime.
It is possible that Joe Exotic is guilty of cruelty to animals.
Under Oklahoma Statutes Title 21, Chapter 67, Section 1685, "Any person who shall willfully or maliciously ... destroy or kill ... any animal in subjugation or captivity, whether wild or tame, and whether belonging to himself or to another ... shall be guilty of a felony."
It's not clear to me whether this constitutes a willful or malicious killing. If I were his attorney, I'd argue that killing that horse was no different than killing any of the other animals that are fed to his tigers. Killing the horse was not done with malice to the horse, but simply to ensure the preservation of the tigers.
Joe Exotic has not committed any civil wrong.
Joe has not breached his contract with Alice. He may never have had a contract to begin with, as a promise to "treat it well and give it a good life" may not be specific enough to have given rise to a binding agreement to do anything in particular. Even if you were to construe it using trade customs, it's not clear that Joe would have had any duty to preserve the life of an already aged and infirm horse.
And even if there were a breach, Alice needs to prove damages, which she probably cannot do. Any emotional distress she endured would not be recoverable as damages, which, in contract cases, are typically limited to economic loss.
She may try instead for infliction of emotional distress, but Oklahoma law would bar that claim, as awards for emotional damages are limited to cases in which they are accompanied by physical injury. See, e.g., Wilson v. Muckala, 303 F.3d 1207, 1213 (10th Cir. 2002) ("Oklahoma law obligated [the plaintiff] to provide proof of some physical injury, whether incurred contemporaneously with her emotional injury, or whether as a direct consequence of her emotional injury.")
Because Alice suffered no physical injury, she cannot collect for emotion injury. Without cognizable damages, she has no viable cause of action against Joe Exotic.