[asid61]

Yes, the robot can't move when the PTO is engaged (like it should).
The PTO is going to a 22t sprocket. I would most likely route it to a 40+t sprocket to lift anything (like the robot) or route it to another sprocket and gearbox next to the mechanism. Given that you would be using 4 cims to lift things, a 12:1 gear reduction should be enough for a linear climber.
I had to design the shifter shaft myself. The idea is that the shifting is on the upper shaft, so whenever a gear is engaged the other gears are not. This also enables me to use only two shafts.
The Vex ballshifter shaft requires at least a 34t gear and uses a 3/4" shaft, plus only has two positions. Using a 3-position 1/2" steel shifter shaft enables the use of a tiny 20t ballshifter gear, and reduces the size of the gearbox considerably.