Based on your flowchart it seems you have done most of the appropriate experimental controls to eliminate bad plasmid preps and “incompetent” BL21(DE3) competent cells as the problem. When you say transformation failed, do you mean that you get no colonies at all? Or you get colonies but they have incorrect (deleted) plasmids?
I’m not an expert with this Novagen plasmid/host system. I’d revisit the Novagen manual and make sure you have the correct Vector/Expression host pairing. I took a quick look and BL21(DE3) is indeed the most commonly used strain, but there is one called BLR (DE3) that is a recA- derivative of BL21(DE3) that may stabilize some target genes with repeats. You also have to consider the possible toxicity of the expressed protein on the bacterial cells. Though that would not explain why you can’t get empty vector into the expression host. Also, when you get stuff from other labs (part A of your chart), you do have to consider the possibility something was not labeled correctly and what you got was not what you think it is. So I’d do some RE digests and make sure the plasmid you are using is what it is supposed to be.
I am baffled, as are you, at experiment B with the commercial construct- 1st generation transformation successful with both hosts, 2nd gen transformation not (but only for the DE3 strain). Did you try the TOP10 cells for that 2nd transformation? That would have been an important comparison.
I hope you figure this out and if you do, let us know!