There is evidence that syphillis came from the Americas. There is evidence it was introduced to the Americas by Europeans. One problem is that it's hard to distinguish teritiary syphillis from tuberculosis or leprosy on bones.
As for Borreliosis, there are many variants of it endemic to Europe and spread by ticks. No evidence that it came from the Americas.
My understanding is that if the tick was only a vector between two humans (for instance, malaria isn't a zoonosis, because the mosquito is just transmitting the disease between humans), then it would not be a zoonosis. But in most cases with Lyme disease, the tick is actually transmitting the disease from another animal to a human, therefore it is indeed a zoonosis.[1]
As for Borreliosis, there are many variants of it endemic to Europe and spread by ticks. No evidence that it came from the Americas.
Neither of those spirochetes are zoonotic.