will mink climb up and run these high walls in culverts?

Yes. But lots of times they will take out before the culvert and go up the bank and cross the road.

I found mink to be surprising jumpers. They can leap up on that ledge and run it. Easy enough to trap the ledge, so you might as well

A mink does what a mink does. He might hit the ledge, the bottom edge or climb out. If I was trapping this I would run lower edge and the ledge at both entrances. And I would would look really hard at the ground around both openings. Around here they really like to get to the culvert, and hunt them and or make a trail going up and around. Also, they will have trails all over, but when they get a pinch or choke point like the entrance they tend to narrow their movement, thus they will have a defined trail. I know I would have at least 8 foot holds and body grips at that site, and stack the mink.

I would also look for Mr. Otter using this site. Old Fathead love places like this.

i would look where the bank meets the tilted footer they should have a well marked trail if they dont i would punch a pocket on both ends of the tunnel and bait with fresh muskrat then if you are seeing sign on the top part
i would use a small body grip with a wire on the trigger but i wouldnt worry about to much fencing to much the mature males will hop or just go around
i would snug the trap to the 90 of the wall either secure it with a coni-mount or use 4 nails in a wide piece if pallet wood and coat either with some mud

If a mink did run that little ledge, it would simply jump any trap set there. As others have pointed out, both ends of that concrete culvert should provide several likely places to make a set or three.