That indicates two things - the amp being quiet when you unplug the cable means the pickups or cable are acting as an antenna. The input grid resistor of Tode is extremely high at 10 Megohms and this can make it very sensitive to stuff that another amp might not pick up. I don't have trouble with it with my guitars, but I don't live near a transmitter either. You might try shielding the guitar pickups. Along with shielding the pickup cavities, on my short scale strat I put a wrap of copper foil around each pickup coil, not a complete wrap, but with a gap that is connected with teflon tape. A full wrap that overlaps will kill the top end. Each wrap is tied to the other cavity shielding with a soldered wire. That guitar is super quiet for a single coil guitar and still has a good top end.
Cables could certainly make a difference too, though my Todes have been quiet with everything from $5 Monoprice cables to the truly great sounding Shunyata Pro guitar cable.
The buzz - from my own experience I will guess this is about your power connections. At the office where we have relatively recent, up to spec wiring a Tode is super quiet. At home where our power grounding is ancient and needing repair the amp does buzz a little bit. Try using the amp at different places and see if the buzz is different.
