In answer to your first question, yes, you can have DT crowntail bettas. DT is just having two lobes, CT is a fin type.
Being that he is a cull, he probably doesn't have a DT but rather has a split tail (misgrown anal fin) that makes it appear to be a DT. Having a split tail would be a culling fault in a breeding program.
That being said, without seeing a picture i couldn't say for sure.
As for why breed culls, it really depends on if the fish was a fault-cull or just an unwanted fish.
For instance, I'll call any fish that I can't/won't use in my breeding program a cull. These include some rather lovely fish. Particularily I have a BF line that throws opaques. I don't breed opaques, so they are culled from my breeding program. Generally I sell them off as pet fish to various people on the cheap. I don't destroy fish unless they have serious problems, such as a deformed spine, etc. Color washes (blue irridesance in a red fish) are another common reason to cull fish from a breeding line.