It looks like a wheel spacer rather than a wide enough rear axle,however I guess it doesn't matter so much,so long as both rims can fit inside on the hub center,before it tappers.Zooming in,it does look like they do.
I've driven with flat tires on the inside to drive to a tire shop without problems a few times.No vibrations but I know to drive her easy until I get it fixed.The whole sidewall blowing out like that,that has to be caused by heat/tires rubbing no? I suspect you had loose lug nuts for whatever reason,long before the flat.