Can an equestrian rider sue a mtn biker if they get thrown because the biker would not approach slowly or yield on a multi use trail?

Can you prove the bike rider's actions caused you to get thrown? You're not going to win a "he said, she said" argument in court. If you have witnesses that are willing to take an uncompensated day off work to testify in your favor you may have a case. You may also have a case if you record your rides with something like a GoPro sports camera and it clearly shows the event and the person responsible. Can you identify the bike rider? If you don't know their name and address then you can't have them served with a subpoena to appear in court. Since this is a civil matter the police won't help you find the person responsible that's all on you if you drag the wrong person to court they could sue you for any costs they incurred to deal with your unfounded accusation.

You could attempt to sue.

However no judge would hear your case. There aren't any laws that were violated and the sign you posted above is a suggestion about courtesy, not a requirement.

Additionally you brought your horse into a public area without the ability to control it. You'd be unable to prove that the fault wasn't your own.

Yes but you would likely not win.

yes , sue the pants off of him/her

Sure.