Conditions on $v \in \mathbb{C}^{n}$ such that the complex to real DFT is sensible
Obviously any operation is allowed that keeps the symmetry of the frequency domain data intact, i.e. any operation that doesn't add an imaginary part to the time domain sequence. And this is the case for all standard filtering operations unless you want to generate a complex valued (time-domain) signal by some operation. This could be the case if you want to generate an analytic signal, but in that case you actually make two signals out of one (the real part and the imaginary part in the time domain). So for normal low pass, band pass, high pass etc. filtering you don't need to worry. As long as you do things right, the symmetry of the time domain sequence should not be changed by these operations.