I have taken mine to nursing homes, schools, Headstart, fairs.
Mine does not poop when he is working, so that has never been an issue. I take a plastic grocery bag, just in case. Use as glove/disposable bag.
Do not visit during lunch hours and do not let anyone offer snacks to your horse. Invitation to nipping!
Slippery linoleum floors can be a problem, so walk carefully or put some vet wrap around the hooves. Watch out for glass exit doors! My horse ran into one the first time.
It is my experience that many nursing home residents do not like animals in their space, especially in the lunchroom. Twice we visited and were taken to the lunchroom, and many of the residents did not like an animal in their dining room. Better to meet in the lounge, or outdoors, though we visited some bed ridden residents in their rooms and that worked out well. Some people do not want their photo taken, so be aware and get permission if you plan on photos.
Children are the most fun, I think. And there is opportunity for education, such as all the things that the letter H stands for.
Do not let the children sit on your horse. Even though they are young, you will get an over-sized child and it will hurt his feelings if he cannot sit on the horsie also. Just don't let anyone set on him, then you'll lessen the problem. Take brushes and combs. Invite them to smell the horse, look at his teeth, feel the hardness of hooves. Show them where the tail bone is.
Children generally want to know if it is a boy horse or a girl horse. If mine is dropped, I am matter-of-fact that he is a boy horse. I don't encourage silliness about that.
My avatar horse can be very ornery. But somehow, when he is with the children and particularly the old folks, he is very gentle. I think he enjoys the attention.
I have a horse trailer, so that is how I haul horse. Sometimes we give cart rides, too. At one time I called our insurance company about coverage, and that is something that is up in the air. I guess I just quit worrying about it.