Is the seriel not a voltage divider to convert the (RS232 15 volt) to the input level as the PIC can handle 5v
The serial input is often only at 5V level from the likes of the serial to USB adapters, so a potential divider as you describe would stop it working, the input logic high would only be 1.33V.
There are diodes on the PICAXE input pins which act to clamp the input pins at no more than about VCC\VDD + 0.2v, the 22K resistor is needed to limit the current ........
The 10K act as a pull down to ensure the PICAXE does not switch to program download mode with the lead out.
I have had many occurances of the "Hardware not found" message, and they have all been errors on my part, normally in the wiring.
The rather simple 'serial interface" of the two resistors works surprisingly well.