If you are looking for great trail riding, Big South Fork in Jamestown, Fentress County, TN is awesome for it. The park itself has over 200 miles of trails, plus there are also some private areas to ride in that are available by purchasing in certain developments or by paying for a pass. Oneida is just across the Gorge in Scott County, and real estate there is a little less expensive than Fentress County. Scott County has the horse developments, so your neighbors there would be horse oriented and able to help you in case of an emergency, etc. I don’t recommend crossing the Gorge with a horse trailer, although there are people who do it. There is a round about way to go to get to trails, but it would be longer than if you lived in Fentress County.
The Big South Fork area has a few Facebook pages with local information on them and some of the realtors advertise there as well. Extensive veterinary care is available about an hour and a half away at UT Knoxville. There are vet clinics with both small and large animal care available locally, but for bigger needs you would have to haul out.
As far as weather, TN has tornadoes. They are just a fact of life certain times of year. Because of this, we chose to buy a house with a basement. You can also find homes with storm shelters. We have a weather radio that will wake the dead when it goes off, that can give you a little forewarning, but there are no guarantees. We also use the NOAA weather app, which is always loud. We had a tornado hit a mile from us this Spring and destroy at least 7 houses and barns, and knock out power and cell service. But, because the tornado took out power lines and cell towers, after the initial warning we knew nothing.
Winter weather in Jamestown can have some snow, but usually not more than a few inches and it is gone in a couple of days. Many people ride all winter. I live off the Plateau, and have experienced winters that have some nights that drop below freezing and will freeze the water trough. Most days are above freezing.
We did contact a real estate agent before we came down to look, but were not super successful with him. I was very specific in what we wanted, and he did not seem to hear or understand what we wanted. We ended up searching Realtor.com, and coming up with a list of properties that we drove by. Out of those properties we found one that we thought was it. We then found a realtor to take us to see it. Because she was local, we had her show us similar properties in the same area, and then made our choice.
My recommendation would be to come down for a week, drive around and see the area. Find an area that suits your needs, then find a realtor.