If I were driving someone else car and got a ticket for having no insurance but later finds out they do have insurance can I go to court and?

That's why courtrooms were invented, in fact, so do that first thing you said. Forget the second thing, because that idea is just stupid. Go with the first thing.

You can even do it before the trial, by making an appointment to meet with the prosecutor. You'll want to bring a copy of the whole policy with you, everything you can get, plus the little proof of insurance card. Maybe even get the owner to request a signed letter from their agent / broker, confirming that the policy was in force the day you were charged. Your explanation to the prosecutor is simple: You couldn't find the insurance card that day, because it's not your vehicle, but you've since located it and here's all the proof.

The prosecutor might dismiss the charge completely, or it might be reduced to something like failure to show proof of insurance (which is a much smaller conviction). But I can't see the prosecutor going to trial and trying to convince a judge that you were guilty when you've got proof that you were not.

You can also wait for the trial date, but it's actually easier and much less time consuming to meet with the prosecutor in advance.

Usually if you go to court and can prove that the vehicle you were driving was insured at the time you were issued the citation the judge will drop the charge. If you were stopped for some other violation and issued a citation for that you will still be responsible for that though.

Yes you can. But then it would be up to the presiding judge whether or not to potentially reduce the fine or throw it out all together.