Nielsen device Q's

It's a modified recoil booster, much the same as on an MG-42 or Vickers, which uses the gas pressure travelling through the suppressor to drive a cup rearwards. It can move entirely inside the suppressor, and all it's trying to do is increase the rearward thrust enough to make up for the extra weight of the suppressor on a recoil-operated firearm.

That's right; it's usually a piston of some sort that's driven against a spring. This allows a recoil-operated pistol to operate the way it was designed to, since merely hanging a pound or more of weight onto the barrel of most pistols is going to make it a one-shot handgun. That mass coming back counteracts the weight of the suppressor, and the spring resets it for the next shot.

Here are some pics of the piston/spring assembly on an AAC Evolution 9 which has an internal recoil booster. The piston protrudes through the end cap and threads onto the barrel. At rest spring pushes the piston away from the cap and holds it against the baffle compartment. Recoil from the gun firing pushes the gun back away from the can, the piston is threaded to the barrel so it moves back with the gun, while the can hangs there. When the recoil is finished the spring pushes the piston back to it resting state against the baffle compartment. At least that is what I can surmise, but I am a newb regarding these things and physics in general.

1) Yes, you could push the piston through the endcap like that.
2) No, the side "lugs" don't slide in tracks, they ride in a mostly cylindrical space. On some cans (I think including the AAC, my YHM has this feature) the lugs can engage a couple studs/protrusions inside the can. Putting different sets of lugs onto those studs when assembling the can will allow moderate POI adjustment.
3) The springs are fairly stiff, but you can compress them a bit.

What is it for? To increase recoil to allow semi-autos to cycle?

It's basically a "decoupling" device. It allows the barrel and slide to recoil semi-independently from the weight of the suppressor. This negates the dampening effect of the suppressor's weight, which would otherwise prevent good cycling.

Yeah the spring is stiff and very difficult to compress. Conquerer is right, it does not ride on tracks, just freely in the space. There is a pin that will set between notches on the piston to keep it in place (you can see it in one of these pics below). You can also adjust point of impact by pulling the can away from the gun and turning it, you can see what looks like a feed ramp in there that must have something to do with the point of impact. Pulling the can away and turning it, locks the pin between a different set of notches and changes the orientation of the can to the gun.