I don't think packing popcorn will work. I'd think it would compress and leave at best a spongy spot in your yard. Most likely you'll end up with it collapsing as soon as any weight is applied over it. The material shown in the two products you listed is a polystyrene aggregate. I don't think packing peanuts are the same exact material.
You could certainly perforate the lower section of the sump basin with small holes (1/4 inch/6mm or less would be my preference, but some might go twice/3 times as large - depends in part what you are bedding it in)
Outside the basin, you'd want washed stone (depending on soil type, possibly filter fabric and washed stone)
Inside the basin, a concrete paver is a suitable pump support. Don't put anything else in there that would make cleaning more difficult.
In my personal experience, pick a submersible sump pump - all the ones I've met have been much more reliable than the pedestal style pumps. To get the most out of whatever pump you buy, don't cheap out on the pipe size - plastic pipe is cheap, no good reason to reduce 2 inch threads to 1-1/2 inch pipe, but I see it all the time - even 1 inch pipe; don't do that - stick with whatever size the outlet on the pump is for best efficiency. And make sure it discharges far enough away from the building - I've met quite a few that pump the same water in circles, from inside to outside, where it comes back inside, to be pumped back outside...
Also, run the pipe straight up as high as it needs to go from the pump, then slope it towards the drain at 1/16 to 1/8 of an inch per foot (or 10mm per meter.)
If you are concerned about pumping rate, you can either try to come up with some sort of test flow (difficult unless the weather cooperates) or simply mount the pump to one side, so that you can add a second pump a few inches higher (use two pavers) if the first proves inadequate when the rains come. If you already have a pump in the old sump, start with that and see how it does.
Best Answer
Those pipes can usually stand to be buried pretty deep. but either way There would be no harm in removing some of the sand and replacng it with crushed rock or pebbles.