the instructor's demo dog was always crated - out the dog would come (minature poodle), and demonstrate some tricks, then back in the crate it would go.
fwiw my preference over time has evolved to have the demo dog crated when not being actively employed.
It's less distraction for the students and their dogs (some of whom may be reactive or overly playful or otherwise easily distracted), and it's less distraction for the handler too. Even in my advanced competition classes, the demo dogs are in an X-pen or crated when not actively working.
As for what breed: I was going to say "it doesn't matter" but on reflection, that's not true. I do tend to expect my agility instructors to have one of the Seriously Serious agility breeds, and for my obedience instructors to trot out a Seriously Serious obedience breed.
It's been a while since I took a pet class, but I recall thinking more of the instructor if they had an obvious shelter mutt or a rescue pittie, because those dogs aren't always the easiest to work with, and if you can get one to do basic manners, then that's pretty good.
OTOH when I was taking pet classes, if the instructor had a doodle, that instantly dropped their credibility in my eyes. I just expect "dog people" to know better. And yes, I know that doodles can be adopted out of shelters or rescues too... but I also know exactly how rare those guys are. (I have an incoming foster that's a Wheaten/Yorkie mix, and she's getting serious adoption inquiries from Canada. And she's not even here yet, and
she has a known [and disclosed] health issue! That's
how rare those dogs are.)