I haven't heard of the trains stopping for snow anywhere in Kyushu, but the higher elevations definitely get snow, even on the coasts. There is even a ski slope to the west of Takachiho in Gokasho. When it snows down low it is generally wet and can stop traffic until the sun comes out and melts it off. The temperatures can drop below 0°C at night but daytime temps are almost always above 0°C. Takachiho can be cold, being in the mountains at a higher elevation.
That said, people go up Mt. Aso and drive the Yamanami Highway in winter when the weather permits. You may lose a day or so if the weather doesn't behave, however, especiallly in January and February.