afak there are non walmart mongoose bikes and they can be quite nice, but most come from wally these days.
The chain might skip because of frozen links or being too loose. If it has been sitting, it is likely that it never saw avid use, so the chain may not be stretched. To check if the chain is stretched, simply attempt to pull a link away from one of the chainrings (front gears). If it comes all the way off the tooth, it is badly stretched, though if it moves more than a few millimeters it is suspect. They make a tool/guage for a more precise reading, but this method works well enough. To check if the links are frozen or sticky, you can watch the motion of the chain while backpedaling the bike. A more accurate check would be to individually attempt to wiggle each link though its normal range of motion and see if any catch.
I guess the more relevant concern is the condition of the freewheel. The rust is often superficial and can be brushed off with the aid of a solvent. The greater concern is that the teeth are not too worn. If the teeth are completely flat on the top, they are in good shape, but if they look like waves cresting or jagged points, then the freewheel probably needs to be replaced. Maybe someone will come by and post a picture for you.