Which manual are you looking in?
See manual 1 pages 25 onwards.
The 22k and 10k resistors for the input circuit for programming the PICAXE chips. The same circuit is used for all PICAXE chips.
If you move the PICAXE to your own PCB where you do not intend to program the chip, then you MUST ALWAYS retain the 10k resistor to ground .
Otherwise the PICAXE will "think" you are trying to download a new program and stop executing the installed program.
EDIT:
my guess as to the "whats this, whats it for":
likely is a poly type capacitor which is used to help filter out higher frequency spikes casued by the switching of transistors within the ICs. Often called a decoupling capacitor.
The Electro:
(I don't have that AXE020 board) but likely for power supply "support" so when you switch loads with the L293D or ULN2803.
In simple laymans terms, the Electro type cacacitor acts as a reserve to help stabilise the voltage where the battery alone might dip in voltage with the load surge. This dips could cause the PICAXE to reset and thus you project would not world as wanted.