So I've noticed that pros (I'll use right handed ones for this example) tend to move right for left hand spares. But I also see them move right for right side pickups like the 3 and 6 pin. Why do they do this?

Does anyone else move right for right side spares (besides he 10 pin)
And if so, what are the board adjustments you make?

I move right (21st board) for 3 6, 3 10, or 3 6 10 leaves and for the 3 or 9 pin. I stand on 35 for my 10 pin and 30 for the 6. But I also stand on the same spot for the 3 6 combo which is a heel shift of 1/2 a board to get the 6 pin. so I have two ways of getting the 6 pin.

I struggled in the first half of this season with any of the combo's involving these pins, 3 6 10. I normally stand left and use the "4th arrow system" which somehow didn't work for me anymore. Now I stand on 21, finish on the 21st board at the foul line and go up the 3rd arrow. And depending of what combo I leave, I adjust my feet by shifting my heel in/left by 1/2 and shift my target right 1.

I do something similar with 2 pin combos - buckets, 2 4, etc... I stand left. For the 4 7 pin leaves, I stand right.

This adjustment, after trial and error, really improved my multi pin spare percentage.

Dont focus on where they stand, go and practice throwing a straight ball at all spares minus the double wood leaves, do this until you can throw straight at them from anywhere on the lane.
Once you commit to this and can accomplish it, straight is straight no matter where you stand on the lane
this is usually best if you learn and practice the proper fundamentals.
Once you commit to that, and can do it, it becomes much easier