Does anyone leave traps hanging in trees or stashed after season, then reset next season?

Yes, just about all of them.

I have done that. On a walk-in line I used to run, I had multiple traps hung/stashed at each set. Made swapping out traps with frozen marten easy. Coni's, of course.
Would rotate traps out in spring, along with line maintenance. Packing bait was work enough. Packing traps on top of that lasted exactly one year.

I'd be leery of it in this country with the price of wolf traps. Deer and elk hunters (and shed hunters) want every wolf gone, but they act like any traps they find in the woods are free for the taking. And that's when they're set, I can only imagine what they'd act like if they found them stashed.