are jack russell terriers born with short tails
No, long tail.
Jack Russell Terrier
They typically have a muscular body with patches and a long tail. 
wiki:19561868
Jack Russell Terrier [SEP] Crossbreeds
The planned-mating crossbreed of a Jack Russell terrier and a beagle is called a jackabee. Jackabees are believed to have first been bred in the United States. They typically have a muscular body with patches and a long tail. The breed brings together the hunting terrier with the scent hound.