Hello,
Vinst the cohort's first power is to return him to the box to "automatically close" the location you're attempting to close. Do I read this correctly in that you apply the "when permanently closed" effect but skip the "when closing" part altogether? Like, can I use Vinst to avoid the damage from Torture Chamber or to avoid banishing a Corrupted card at the Sanctum, etc. ?
Yes. Boxing Vinst closes the location without needing to do the When Closing. :) So it can be worth it to "blow" your first explore to pick him up.
Are you allowed to keep using him throughout the scenario?
Each time you send him to the box, you can use your first explore to get him back, so yes.
(Note: Not in B-5 though; he shows up there, but there's no mechanism for getting him back in that scenario.)