1. Should AD have checked gun then Alec also checked gun?
The armour should have prepared it, and then the AD checked. I wouldn't expect an actor to necessarily know, on an old gun as was used where the bullets are in an external chamber, perhaps a quick eye over it but when an expert is in place and it's double checked, it's not standard for actors to check.
2. Should it have been aimed at the director/cinematographer at all when direct to camera shot?
They do usually retreat to a viewing area once the cameras are set up, so in theory there shouldn't be anyone in the vicinity for direct camera shots. A report says it discharged when he removed it from the holster, rather than it being to capture a shot; which would explain why this wasn't the case.
3. Should a protective screen not have been used?
No, usually there wouldn't be anyone in the field of a planned shot, but it seems this was accidental discharge, something they will hopefully tighten regulations for now ie cameras set up and everyone moved to the secondary area before the gun is handed over.
It will all come out eventually.
Yeah, there were a lot of witnesses and it seems people are cooperating so I'm sure it will.