Very broad strokes discussion here…
Civil cases are brought between private parties. If a person has been wronged in a way that the law recognizes as a civil cause of action, that person can sue. The person can also elect not to sue. All of us have probably had experiences that gave rise to a civil claim and we decided just to move on with life. There is generally no obligation on any person to file a civil claim.
Most civil actions operate under the “American Rule” where each party bears his/her own costs. So if P sues D for tort and wins, P still has to pay P’s own attorney. This is one reason why we have contingency fees in a lot of cases, the attorney is only paid if the attorney’s client prevails and out of the proceeds of the judgment.
There are exceptions to the American Rule. By statute, some civil actions permit fee shifting. One example is Title VII. If P (employee) sues D (employer) for employment discrimination and P prevails, D can be ordered to pay P’s attorney’s fees (and D would also pay D’s fees). In addition, parties can agree to fee shifting in a contract. That may or may not be enforceable. It is an exception to the usual American Rule, which is the default rule in civil cases.
There are many prerequisites for a contract to be enforceable. Among other requirements, the parties must have a “meeting of the minds” and agree on the terms and there must also be consideration (something of value exchanged or a change in position). There are many reasons why what otherwise seems like an enforceable contract might not be. For example, if one party lacks capacity (is a minor) the contract is not enforceable. Another example is misrepresentation. For example, if someone made representations about their breeding program, the condition in which they kept horses, etc., those could be construed as material misrepresentations that rendered an otherwise enforceable contract unenforceable.
Assuming there is an enforceable contract and a breach, then the question becomes what damages are available for breach. Compensatory damages are the typical form of damages, whatever money is necessary to make a prevailing party whole. If I paid $4000 for a horse and spent $50 getting a halter with the horse’s name on the plate, and then the seller backs out-- I could sue for the purchase price and possibly the cost of the halter. Whether I could get damages for the special damages would turn on various factors. If the halter can have the nameplate changed for $5 I am probably not going to recover the $50, I am probably going to recover $5 because I haven’t entirely lost the value of the halter.
Specific performance is another form of damages but it is generally only available where money damages are insufficient. If you agree to sell me a one-of-a-kind priceless painting, that might be a case where specific performance might be ordered-- you might be required to give me the painting. This is pretty unlikely in a case over a horse because you can have the value of the horse assessed and then take that money and buy a replacement horse. In most cases horses are going to be viewed as interchangeable chattel that have a value which can be monetarily valued.
There are other forms of damages as well (none really being discussed here).
Damages will be reduced by any failure to mitigate by the plaintiff. If the plaintiff is offered a refund that she could use to go buy another horse and she chooses not to do that, she doesn’t get to claim additional damages because she didn’t have a mare available to breed this year. All plaintiffs are expected to mitigate their damages within reason. Someone who has been offered a refund and doesn’t take it faces a risk that s/he will be deemed to have failed to mitigate damages that follow the offer of a refund.
A party who has been sued might be able to assert a counter claim. Some states recognize a tort of “malicious prosecution.” This is defined differently in different states, but generally it involves the filing of a lawsuit for an improper purpose. So going scorched earth on someone who backed out of a contract and offered to send you a refund of your purchase price as revenge for what other people posted on social media could arguably meet this standard.