There were cats in Plantagenet England. A commonplace you might think, amd yet, rabbits were sparse in that place and time. In fact, rabbits are said to have been absent from Ireland up to a mere century before the beginning of what we call Plantagenet England (1225 to 1485). Michael Prestwich, a professor of history at Durham University since 1986, and author of Plantagenet England 1225-1360 (2007) notes that at the time in question, the only"rabbits were carefully looked after in warrens."
One reason for the caution was the threat of wolves, who were in remote areas a vivid threat. The Earl of Lincoln's estates in Lancashire lost 11 head of cattle in 1304-4, to wolves. At this time England's main non-human animal was sheep. "England was a country of sheep." (This would continue many centuries). Rats were not that common, despite their role in the plague turnoil. Barns at the time were pretty open, and stored the harvests with rare indications that mice or rats took a share.
Professor Prestwich in fact does not mention the value of cats for pest control, (though earlier the Welsh had valued them for this reason, an indication of feline rarity.) He categorizes cats as pets. We get a nice vignette in his text: "John de Engayne was accused of catching a domestic cat belonging to Millicent de Mohaut when [he was] hunting wolves, foxes, and wildcats."
And there were horses, or not, if your name was Richard III, and your biographer Shakespeare. In fact Richard's death on August 22, 1485, marked the end of the War of the Roses, the end of Plantagenet contention for the throne, and, according to some, the end of medieval England. It was not that the Tudors won. It was that others with a claim to the throne were mainly dead by then. This man who could not stand up straight, fought bravely at the Battle of Bosworth Field, and at the end they stuffed his body into a small space. And forgot where he was, for over 500 years. Until his remains were found under a parking lot in 2012.