If the bucket not being level had anything to do with the hydraulic fluid system it would have gone away just by hitting top and bottom of lift travel a couple of times. Something is bent or shifted on that one. The fact that it goes level at the top and bottom is just the loader hydraulics flexing it back the other way, but not flexing it past its 'yield strength', so when you release it just springs back to 'bent'. With careful setup you might be able to use the loader hydraulics to 'unbend' it just like people do with their SSQA cross-tubes.
The bucket rocking so easily while the valve is in the 'neutral'/middle position likewise COULD be air but that should have rectified itself just by fully extending and retracting the cylinders several times. What makes me think it still IS air in spite of that theory, is how easily it moves. Even if a piston seal or a spool valve leaked, you still have to 'pump' fluid through that 'orifice' to get the cylinder to move, and basically ALL of the fluid in that circuit has to move to do it. Even without much restriction from the leak/orifice, it's still a fair amount of inertia and more like stirring honey than just flopping around like in that video. If you consider something like leaving the bucket tipped down a little, putting the loader in float and letting it hit the ground, turn off the tractor, and then pull the bucket curl valve, the bucket will 'settle' to the ground much slower than the movement shown in that video, and that's with the valve fully open with the whole weight of the loader and part of the weight of the bucket on it, not just something 'leaking' with one measly boot pushing down on it. The system has a LOT of air in it!
As for how the air got in there and then.. never left or never stopped reappearing, i'd be especially suspicious of shaft seals on the cylinder glands, valve body handles/levers or at the pump. A lot of hydraulic pressure seals are directional lip seals, which means their ability to seal goes up with the pressure pushing them against the shaft, and down when that pressure goes down. They're also bad at resisting any pressure from the 'outside', which is usually just atmospheric pressure so not usually a problem.
But think of this. This loader works when the tractor runs. When you pressurize circuits, it more or less becomes normal. Once you fully relieve the system pressure, it becomes wonky and may only do so after sitting for a while. My theory is that a directional lip seal is now allowing air to suck into the system when the pressure drops to 0, but then seals again as soon as you MAKE pressure in the system. I have seen something similar to this on leaky cylinder gland seals, as im sure many others have. They seal 'better' when retracting the cylinder because pressure is forcing them against the rod, but leak profusely upon extension because no real pressure builds in that chamber when the control valve is open on that side.
Just because hydraulic pressure drops to 0 doesn't mean there's nothing to generate 'suction' in the hydraulic system. The weight of any hydraulic fluid being held above the sump that has a free path to flow back to the sump is only being held in the lines by the ability of the system to keep air out of shaft seals. If you picture sticking a straw in a drink, covering the end with your finger and picking up the straw, the liquid only stays in the straw because your finger is sealing against that 'suction'. So my perhaps left field theory is that you might have damaged a directional lip seal which is allowing the weight of the fluid held above the sump to suck air into the system every time the system pressure drops to 0. The loader itself never moves because it's already on the ground, only the fluid in the system moves as some migrates back to the sump and is replaced with air. Once you start the tractor, the pressure causes damaged seal to seal again, and loader will more or less immediately work, although you might notice its initial response to the first lever inputs is delayed (squeezing the air pockets before making real pressure), and its ability to hold weight is 'springy' (air spring) until you've chased all that air back to the reservoir and purged it, at which point it becomes 'solid' again. Until it's parked for a while!