Using DFT Rotation for Frequency Down-Mixing
Is it possible to perform Frequency Down-Mixing ... by simply rotating the DFT sequence?

Yes.

If so what are the Advantages and Disadvantages of this method?

As compared to the standard method of just multiplying with a cosine or complex exponential, there are no advantages (hence, no one really does it). There are lot of disadvantages: it requires framing including overlap management and potentially windowing, it's computational expensive and calculates a lot of stuff that you actually don't need.