I can help with headrests maybe. Do you hear a motor running when you try to raise or lower headrest, but nothing moves? If so, that's good. The spiral cable that runs between the headrest motor and the headrest gear actuator is notorious for shrinking in length after a few uses and the square spline at the end of the cable then slips out of the gear actuator mechanism.
You have to pull the back cover off the seat and you'll see the black shielded drive cable running from the motor to the headrest actuator gear. The easy fix I used was to cut the shielding in the middle of the cable using a razor, then remove about a quarter or half inch of the shielding (cutting the shielding twice). Top end of the cable inside the shielding can be pulled out after the first cut...it'll be greasy, so minimize handling of it and use a towel to keep grease stains a minimum.
Once you've removed the small piece of shielding, get a short length of 3/8" fuel line and a couple pipe clamps. Slide the fuel line and two clamps on to the shielding, reinsert the spiral drive shaft into the empty shield and then push the two ends of the shielding back together. While holding the shielding ends together, slide the fuel line over the splice, and clamp in place centered using the two pipe clamps.
This allows the spline to push the extra 1/4 to 1/2" back up into the headrest gear and if you give it a try, you'll find it works now just fine. You can wire tie the cable back in place against the seat back if necessary and then just replace the rear cover of the seat back in place.
If there's no motor noise on yours right now when you try to raise and lower, then most likely its an electrical disconnection. But typically, the problem is the short spline that just needs to be pushed back up and held in place. Shortening the shielding solves that problem and you should be good to go.
Not sure about the tilting seat thing, but have seen others have that problem...hopefully someone posts about it for you. It may be a similar issue as the headrest, but I didn't have that problem myself. Not even sure if there's two motors or one, but it has something to do with both side gears not actuating...think you can see them with seat back off and hopefully get better idea of why seat tilts when rasing. I think you can wrestle the seat back square by hand while actuating the mechanism, but that's not a fix. Good luck with that.
Lastly...have no idea what the mystery mechanism is...can't tell from pic. Never took my seats out either. Wish my armrests were in as good a shape as yours...trying to find reasonbly priced leather covers to replace mine. Seats are good, but armrests shot. Otherwise, your interior looks identical to my '94 530i. I even have the same black and yellow screwdriver set
Best wishes on the seat repairs. I don't adjust my headrests almost ever, but nice to know they work now.