The original dog playing Beethoven was only in the first two movies. He is no longer alive but some of the dogs in the later films likely are. His real name was Chris, owned and trained by Karl Lewis Miller who also trained animals for Cujo, K-9, Babe, and many others.
I have met the owners of the Beethoven who played in Beethoven's Big Break. They have amazing dogs. They are in Western, NY called Johnson Farms