Yes, the bacterial cell body rotates, but more slowly than you might think.
Bacteria are examples of 'life at low Reynold's number'. Here are some extracts from Howard C. Berg Random Walks in Biology (1993) Princeton University Press pp 75-80.
The Reynolds number is a dimensionless parameter in the equations of motion of a fluid that indicates the relative size of terms that describe intertial forces (forces required to accelerate masses) and viscous forces (forces due to viscous shear).
For a fish R is approximately 105; for a bacterium it is approximately 10-5. This difference has a profound effect on the physics of swimming.
The fish propels itself by accelerating water, the bacterium by using viscous shear.
To illustrate the lack of inertial effects at low Reynolds number, Berg estimates that if a bacterium stops swimming it comes to rest within about 1 µs and coasts for approximately 0.4 nm.
Flagellated bacteria swim by rotating one or more thin helical filaments... Torque generated by rotation of the filaments is balanced by viscous drag due to the counter-rotation of the cell and thrust generated by rotation of the filaments is balanced by viscous drag due to translation of the body of the cell.
In a recent paper from Berg's group direct measurements of the rotation of the flagellar filament and the cell body (for E. coli) are presented: the filament rotates at approximately 100 Hz; the cell body at 20 Hz.
The supplementary question is: Shouldn't the rotation disturb sensing and nevigation in the environment by the cell? Bacterial chemotaxis, at least in E. coli, relies upon time-averaged sensing of the concentrations of attractants in the surroundings via ligand-receptor interactions. Rotation of the cell body will not have any effect on these interactions.