So if I'm looking at that right, the ball valve with the red handle should be a on/off, or engage/disengage... and the little twist valve that looks like your ice maker water hookup is the control for how fast the fluid can move. So all that thing is doing is moving the fluid from one side of a plunger to the other, as the cylinder goes through its stroke? Can you take it completely off to R&R? And is that hose on the left completely crimped, or is there still pretty free flow in there?
If its behavior changes mid stroke, then yeah I'd speculate you still have some air in there. Is there a way to crack the whole thing apart, and see if there's an o-ring inside that could be replaced? Even though that wouldn't really solve your problem, as you should be able to compensate for a bad internal seal with just more dampening at the valve, so it sounds like it's leaking somewhere and sucking in air. You could try the soapy water thing, or just T-tape up all the joints really well and see if that solves it.
Another way to test for leaks is dump the oil then fill it with water- water is less viscous and would reveal your leak sooner. Or go the other direction and try putting some super thick fluid in it, just to mask whatever isn't working right.
How long is it compressed, and how much stroke does it have? You may be able to just pick up a spare and not even mess with it..
And if I were to totally re-engineer the thing, what if you replaced it with an air cylinder you could activate with a foot pedal, hooked to your shop air. Give it dampening on the way down, but hands free on the way back up!