My reading of the schematic is that the Launchpad pin functions are as follows:-
1 +3V supply to micro,
2 analogue input, monitoring photocell,
3 analogue input, monitoring battery voltage,
4 no connection,
5 digital input, monitoring coop door switches,
6 digital output driving Q1,
7 digital output driving Q2,
8 digital output driving Q3,
9 digital output driving Q4,
10 GND
Q1 and Q2 are both for motor control.
Q3 and Q4 respectively set and reset a two-coil latching relay which operates switch K1 to connect/disconnect the solar panel.
What actual logic the Launchpad applies to its various input and output signals is not altogether clear. Clearly you would have to replicate the overall function of that with the Picaxe, as well as replicate all the I/O connections.
The Launchpad checks the light level and if it is above some threshold opens the coop door. It closes the door when the light level drops below another threshold.