I have known many ex-racing greyhounds, and have trained a few to learn the ins and outs of being a pet, not to mention tricks and other fun behaviors. So they are definitely trainable.
A lot of what Gypsy did in the film looked like fairly natural reactions that they might have gotten pretty easily, even with an untrained dog.
Many greyhounds are very responsive to stuffed toys like the one Lana Turner's character brings Gypsy, and/or any toy that makes a squeaky noise.
And while they are not generally big noisemakers, some are, and seeing anything exciting can get many of them to make a speech. Walking into an adoption kennel to volunteer, I was always greeted with a chorus of rooing and barking.
I don't know if Gypsy was actually watching other dogs race or if they got that reaction some other way (at the moment, I can't remember whether we can see what he is looking at when he barks, or if there is a change of shot that makes us assume he's watching the race).
And I must force myself to stop, because I could write about greyhounds all day!
I enjoyed seeing the Greyhound a lot in the film. And I'm sure many other fans- like yourself- did also. So I assume Gypsy's role was carefully set up (for want of a better word).
It looked like Gypsy was watching (through a window) other dogs race. Still, to get that kind of barking reaction seems remarkable. I thought it was hilarious.