Does the pump run? If not, figure out why. Dismount the motor from the pump and make sure that's turning. If it does, then doesn't turn when mounted on the pump, the pump itself could be locked up. Yes, you can get water and rust =inside= a hydraulic system and it =can= rust.
Do the solenoids move? Most solenoids I've had experience with have manual activation by pushing on the end of the shaft that sticks out. It appears that yours have this. Give that a try.
I'm assuming that the first pic is of the landing leg motor and cross-shaft, with the very bottom showing part of the hydro system.
If the slide is out, you could remove the hose on the return side of the cylinder(s) at the pump, put them in a bucket, then try manually pushing the slide in.
'Bout all I can think of right now. Given that I've never had an RV with hydro, just grew up farming and dealing with hydro systems on tractors and implements.
Lyle
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