When you get an extra use out of an enchanted item by taking mental stress does that require an standard action?
Mechanically? No. Indeed, it takes no action. Activating the item takes whatever action it normally requires, completely unmodified by the fact you're taking Mental Stress.
Thematically, it represents, at least IMO, the sort of reflexive, possibly desperate, pouring of power into an empty well to make something work which doesn't quite have the juice to do so on it's own. It doesn't even work on items with uses left, and that makes perfect sense in my head. As does the inability to extend it as per Evocation. I know exactly why this makes sense to me, but I'm tired and words aren't coming. Maybe in the morning.
By the way, when you create a Rote that extends duration does it do that for only one designated spell or for any spell it's used on? I don't see it really being game breaking to do the latter except becoming incredibly useful in its utility.
The rules are unclear, so GM call, but I'd allow it. You have a limited number of Rotes after all, and burning one on this means you don't have it for anything else.