Hi s.guo1,
That’s great news. I’ll look forward to your future updates.
Most manufacturers would not recommend that you use an actual coil box spring unit with their product, but instead something that is flat and firm, like a slatted network. All you need to know about appropriate foundations is listed in the foundation thread here.
Most of these mattresses are not designed to be used with a coil box spring and it will actually degrade the performance of the mattress, and also usually violate the warranty (always check with the mattress manufacturer).
Cardboard will not allow the mattress to breathe, at least on the bottom. As long as the slats were closely spaced together (5" or less for most polyfoam mattress cores, closer would be better), you’ll be fine.
Check with the manufacturer regarding bowed slats and see if they want you to avoid those. Keep an eye on those systems, as some use less durable slats and over time the can begin to sag. They will also slightly change the feel of the mattress. There are plenty of options using flat slats.
There is more about mattress protectors in post #89 here and this post. The polyurethane film that most of these protectors use is somewhat breathable, but they fit on the mattress like a fitted sheet and the film is normally only on the top surface, so the mattress overall still can breathe quite well.
Phoenix