The two black screws look like the screws for the oiling ports of the linear bearings. Each bearing has two screws, one at each end.
The other parts are from the anti-backlash nut. Here’s a good thread on how to assemble it.
https://community.carbide3d.com/t/z-plus-leadscrew-nut-replacement-how-to-guide/53601/4
You can run without the anti-backlash parts, but you may find there is ‘slop’ in the vertical movement of the Z carriage. The whole design of the anti-backlash nut of the Z plus is troublesome, it’s not clear that the design actually helps reduce backlash at all.
Finally, the fact that the AB nut fell off, AND you were getting a deep plunge, most likely means you are losing steps when the Z carriage moves upwards. Check lubrication, see if there’s any debris on the rails, and check the motor wiring looking for damage or corrosion, or signs of overheats (which can come from electrical arcing).