I think the black piece should be attached to the metal rod behind the optical unit, to the right side as we see it in the picture. It is probably meant to block the optical unit from travelling too far back and it helps support the metal rod from not moving. I guess that the copper colored screw is there to block the metal rod from lifting itself out. Check the black piece and look for a rectangular hole in the plastic where the screw sits. It may be this part and I think it will be replaceable if you get the part number.

I guess I already see a broken connection in the ribbon cable for the optical unit. Far left on the angled ribbon cable you can see a cut-out, probably caused by the black piece floating around in there and scraping everything it touches. These ribbon cables are fragile and easily damaged. This may cause trouble for the laser optical unit and also explain why it won't work.

Good eyes laserpaal!

Hey guys,
We've put a very small peace of metallic wire between the two exposed points of the cutout, and glued back the piece but nothing happens.
Tray won't open unless we open it and the tray won't go up and disc won't spin (LD, CD, 3" CD single).
Loading tray belt is not round anymore and looks like an egg - is it worth replacing or no hope for this unit?

Maybe it is possible to check for continuity on each of the traces of the ribbon cable, to verify there is continuity and that will also verify if the metallic wire works as a bridge.
The belt is the least expensive part, so it's worth the shot. Can you hear motor movement?

The black plastic piece is the end of the tray on the right hand side (looking from the front).

To my eye, it looks like the pickup isn't properly set. I think the pickup has an anchor point on the left (again looking from the front) where the guide sits on the protruding lip. The pole should hold it down on the right and then you should NOT be able to raise or lower the pickup on the left except with adjustment screws (Tangential Adjustment).