The most probable cause of the symptoms you describe is a leaking flush tank.
The bad smell results from sewer gas leaking past the wax ring, which results from the toilet moving when you sit on it, which results from deterioration and rotting of the floor, which results from the floor being constantly just a little bit wet, which results from a small amount of clean water seeping from the tank.
The intermittent cycling of the fill valve results from a slow lowering of the water level in the tank, which results from a small amount of clean water seeping from the tank.
This kind of leak can be very hard to find. Often there is no dripping and you can detect the seepage only by wiping the outside of the bowl and the surrounding floor with paper towels or toilet paper. The leak can even be completely inaccessible, if the ceramic bowl is cracked under the tank seal.
In a case like this, with the moving toilet and the bad smell and all, you will have to remove the toilet to diagnose the problem.