I would guess double attack can proc during any of valoredge's weaponskills, though I don;t have any proof.
If my new hypothesis pans out, though, we may also be able to get more favorable shield bash performance out of valoredge using the tactical processor.
People tested Shield Bash with Tactical Processor many years ago and found no improvement. Tactical Processor still sucks, and the explanation was wishy-washy anyway. Not worth a slot.
As for DA on String Shredder, IIRC automaton WS work like player WS in that DA can proc on any hit of a multi-hit WS (up to 8 attacks total per round). To confirm, just do a lot of WS and immediately check the puppet's TP. Each DA proc will return an extra 1tp.
Also, I think VE sucks now:
1) SS stomps it for overall DD
2) VE's shield blocking doesn't really do much if it doesn't have hate (and it has poor tools to keep hate)
3) Survivability on automatons is irrelevant due to 1min DEA timer making puppets essentially disposable (for free, so I'm sure that causes some jealousy for BSTs who have a 4:10-5:00 call beast timer AND need to use consumable jugs to call the pet)
Unless they add new JAs/attachments/job adjustments making it practical to want to tank with an automaton, VE doesn't really do anything worth using it over another puppet. I wish VE was good, I want to love the little guy but I simply don't see it in today's FFXI.