I disagree with Gottatrot. And agree with MIE. Basically yes. Most breeds have standards. But you may find a bulldog that LOVES to run. And a lazy border collie..
As for horses. Yes TB's and Arabians are known as "hyper" but I know some VERY mellow ones. So don't limit yourself by breed and gender. Find something, that's a know it all, and quiet. I wouldn't even limit age. We just bought an APHA mare that is 3 years old. That is more quiet and gentle and willing then all of our more seasoned horses, and we had a 17 year old that has been broke since he was a baby, but was still an experienced rider only horse. I say go all out. Look at all see them all, visit them all! You never know which you will "click" with and which are just totally not what you are looking for. But to know this you will actually have to go see them. Just remember the vet check and the knowledgeable horse person :)
Baby, Sparta, Carmen, Henry, Hooch, Mercedes, Butterscotch