It sounds like something is wrong with the motor circuitry, like it is causing a short circuit or to big voltage drop to cause the reboot. The switches K1 and K6 seem to trigger it, but that is probably because the analog converter returns a slightly different value from what is expected. (resistor value tolerance might be the cause, but not neccessarily) K6 represents the "least significant bit" in the PollSwitch value, AKA 1. This one is supposed to be pressed to turn on the left motor. But if the resistors for K1 (or the others) is slightly of, or just because the analog to digital converter is not exactly linear, the result might be an un-even value and then K6 appears to be pressed. But it is not obviously. The motors produce electrical noise by themselves that feeds through Vcc into the switch measurement circuit. (The H-bridge of the motors hangs onto Vcc. Pin PD3 used to feed the voltage divider for switch measurement is internally connected to Vcc when it is measuring. And some disturbances might also feed through R11 into the AVCC and through R23 into the switch measurement pin PC4)
R13 has nothing to do with what happens during the SelfTest. It is only measured during the Bootloader phase. But if the short is allready present at that time then it might make the Statusled flicker, and send out VL (or just V, not sure) over the IR communication.
So I would put the focus on the H-bridges (transistors and diodes) and the IC3 that controls them. I don't have a clear idea how to systematically analyse the problem. I guess the best solution would be to take out the atmega8 chip and simulate the signals it sends by connecting those pins to Vcc and GND with wires. And then measuring the voltages on the pins of the resistors, transistors, and the NAND IC3. But I say that easier than it probably is. I never analysed it myself. So maybe someone else should take the lead here helping you out. (or maybe send a mail to Arexx Customer support).
The programs on this page (from Arexx-Henk) may also be of use identifying the problem exactly (i.e. DiagnoseAsuro ):http://home.kpn.nl/h.van.winkoop/Asuro/ ... PagFrm.htm
(Warning!, It's best not to Flash another program into it (overwriting SelfTest) if you are not yet sure that the IR communication is working properly and reliably. But it seems it does, so you're probably ok.)