I'm sorry! Um, it probably doesn't help that much of the relevant info hasn't been published yet; this patent is the best description that will be publicly available until the inventors get more funding. From the patent:

By replacing the volume of the vasculature (from 5 to 10 percent of the volume of tissues, organs, or whole organisms) with a gas, the vasculature itself becomes a “crush space” that allows stresses to be relieved by plastic deformation at a very small scale. This reduces the domain size of fracturing...

So, pumping the organ full of cool gas (not necessarily oxygen) is done for reasons of cooling the entire tissue at the same time, as well as to prevent fracturing, rather than for biological reasons.

ETA: To answer your last question, persufflation would be done on both cooling and rewarming.