The only reason I use screen on the big iron is to increase the kill area; maybe it would be better explained as, reducing the tracks inside of the jaws.
If you have the pan tension set at 2.5 too 3.5 pounds. I don't think an edge step will fire the trap, even with screen. In fact, I have a tendency to use the lower pan tension with the screen...I don't know if this is critical.
I cannot remember a toe caught critter or pull out last year with any of my coils. I did loose a cat with a #4 dbl long and screen. The increased weight of the dirt and screen made the trigger way too light. I had filed a Miles trigger into the trap but just didn't have the tension high enough. On #3 dbls , which are dang near identical to many #2 coils, I use plugs. With the Miles triggers and wire to reduce the pan "wobble", things work just fine on these traps.
I also have all my coils set up with a gun-latch (night latch) type trigger. I think this to be a good set up with high (relative to the target) pan tensions and maybe even more so with screen and the bigger iron.
I don't think I have any PICs that would be instructive on bedding. Frankly, with the differnces in soil types, I wonder how this could be done. For example most of the soil I trap in defies compaction adequate for a trap cap set up.