This article investigates the structural stability and sensitivity properties of the confined turbulent wake behind an elongated D-shaped cylinder of aspect-ratio 10 at Re = 32 000. The stability analysis is performed by linearising the incompressible Navier-Stokes equations around the numerically computed and the experimentally measured mean flows. We found that the vortex-shedding frequency is very well captured by the leading unstable global mode, especially when the additional turbulent diffusion is modelled in the stability equations by means of a frozen eddy-viscosity approach. The sensitivity maps derived from the computed and the measured mean flows are then compared, showing a good qualitative agreement. The careful inspection of their spatial structure highlights that the highest sensitivity is attained not only across the recirculation bubble but also at the body blunt-edge, where tiny pockets of maximum receptivity are found. The impact of the turbulent diffusion on the obtained results is investigated. Finally, we show how the knowledge of the unstable adjoint global mode of the linearised mean-flow dynamics can be exploited to design an active feedback control of the unsteady turbulent wake, which leads, under the adopted numerical framework, to completely suppress its low-frequency oscillation.