Yes, she owned three ponies. I don’t know if she ever had a horse, but she was given a pony named ‘spunky’ by the film studio at one point and sometimes her pets were included in the films, such as her dogs and her ponies (Spunky can be seen in Curly Top). I don’t think any of the ponies were very well behaved and there was a lot of trouble in training them, but she did ride them anyway.
I know which set you mean, with the photographs of her on a farm. But I can’t find one of her with a Rooster, sorry!