As a horse trainer or riding instructor do you need to carry insurance?
Yes. If you don't have it you could be majorly screwed if there was an accident and you were held responsible.
Can I carry the insurance on a car that is financed in my spouses name if he is no longer able to drive?
No, there is currently no law anywhere in the United States I am aware of nor any state therein that requires doctors, attorneys or dentists to carry malpractice insurance. It is totally up to the that professional whether or not to carry malpractice insurance and what limits they may or may not desire to carry.