Congrats!!
1. Not sure about this. They may or may not make him redo it. Have to wait and see.
2. NEXUS SHOULD help. When he did NEXUS did he have to provide court documents for the charge? If he did, there's a good chance they have them on record and won't hassle him. No guarantees though.
Either way, that's great news for you. Worst case he will just need to provide a new medical/court documents but it shouldn't delay you that much. That being said, best case scenario is you get a DM in the next few days assuming they already have the court docs and the medical isn't an issue (but again not sure if they will request a redo of the medical or not, I have seen some people have to redo it). The good news is in my experience, those with NEXUS do not wait long for a DM because they have already gone through a background check and in those cases I've seen a DM usually within a week of background check (sometimes as fast as next day!) Don't quote me on that, but I would expect it to be fast!