You need the Star diagnostic tool to check the codes for this. I'm pretty sure that only engine codes are available at the OBD-II port under the dash, as required by US law. The rest are available at the port in the under-hood fuse box. The Star diagnostic tool isn't just more expensive than what you have (or what I have). It's obscenely expensive. Something like 3k/year for the license fee for the software, plus the outlay for the hardware.

My climate control unit usually also shows about 13.5, but the actual voltage at the battery is a bit higher at around 14V. Check yours with a voltmeter to ensure that yours is also higher than 13.5V, which really is a minimum charging voltage (but should be fine).

The brake switch can cause problems that occur when you hit don't the pedal. If it were mine, I'd probably change this part, for what it costs. Be sure to get the M-B part for this.