EPDM and similar foams can be glued quite well. You can glue them with contact adhesive (the sort you apply to both surfaces, allow to go tacky, then stick together, no chance of adjustment). This works for large flat areas. Hot glue also sticks them but getting the thin even layer you need for laminating sheets won't be easy as it will set before you've finished.
In both cases the bond is weakest against pealing, so try to avoid a construction that has you pulling against thin edges.
Thin sheets can be cut with a sharp knife, but over about 15mm (1/2") a hot wire cutter or hot knife is better. Short cuts can be done by heating up a craft knife but there are heated knives made for this. EPDM might not smell pleasant, and you can get fumes if too hot, so ensure good ventilation.
All the same techniques can be used for open cell foam as well, and a sandwich construction using stiffer and softer foam can be effective. Upholstery foams come in different firmness ratings. All but the softest should be good here, and will be a great improvement on the built-in one-size-fits-none stuff you have now. Sharp scissors or a knife will go up to about 25mm (1") with an imperfect edge, again, hot wire is better beyond that.
EPDM has a downside - it's heavy. I've comes across it as sound-absorbing foam, and it's noticeably denser than alternatives like EVA.
One other option to consider, depending on how often you'll be repacking, is the packaging foam that moulds in situ to the item you're protecting. It's essentially the same as builders' foam filler, blown into a bag while liquid. There are expensive systems for custom packing, or DIY solutions. Neither is as visually appealing as a well-made cut foam insert.