Representing stereo signal with complex numbers as input to DFT
For your first claim. Yes it's true. You should keep the full N-DFT complex frequency samples as opposed to half, N/2, for the real input case. And also yes, you will retain back the original stereo signal perfectly by an inverse DFT.

Whether this artificially produced complex signal adds any new information that cannot be obtained from individual channels is not very clear, however. At least from the linearity of DFT we can guess that the resultant DFT of the complexed-stereo pair will be exactly given by those individual left/right channel DFTs, and this suggests to me that, you wont be obtaining some interesting benefit here. 

Keep in mind that, the (geometric) spatial relationship between a pair of stereo microphones can be investigated by a joint processing of those channels. However, a DFT alone does not care for any spatial relationship between real and imaginary part of its time domain complex input. Hence I suspect any good use of that for spatial processing.

Yet, in some cases, there may be a computational benefit due to performing a single complex input DFT instead of two real input DFTS. In such cases you are favored to do so.