The "screw" is the trimmer (variable) capacitor. It is used to fine-tune the oscillator. It can be replaced with a small fixed-value capacitor of ~15pF (picofarads). This has been discussed soewhere in this forum recently...
(update:
here and
here).
Not sure what you mean with "towers".
All components on the board are required for the watch to work. This inlcudes driver transistors (black), surface-mounted (=SMD) capacitors (brown), SMD resistors (white), and through-hole-mounted resistor (brown cylinder with colour stripes). The resistor is part of the oscillator.
Then, there's a diode and additional resistor as part of the charging circuit. These are not required if you feed the module directly with 3V and don't want to charge it with the solar cells.
This old thread has some good information and images of the MK I:
http://dwf.nu/viewtopic.php?t=2180