Yes, if just about anything goes wrong with the FFE, there is a DTC code created. However, you are not able to retrieve these codes with a standard OBDII reader, you need an interface that can handle High Speed and Medium Speed CAN signals. You will also need a laptop with Forscan installed. I am quite happy with my OBDLink EX USB interface. Right now it is $50 at amazon. Some people prefer other interfaces, but they do not have automatic HS and MS CAN switching. You must manually flip a switch. Be wary of any interface that is less than $20 as there are many unreliable Chinese clones. To read codes, you can use the free version of Forscan. To do testing and changing parameters, you need to get a Forscan license. I think they are about $10 per year.
There is another way to see DTC codes, however I am not for certain if that method displays all codes. Read about Engineering Test Mode here:
Once you see the DTC codes, you can either use Google or the FFE service manual for more information. Get the service manual here:
I would not trust that dealer anymore. They are jerks. $2000 to replace an A/C compressor is too much, and it probably wasn't necessary. When they did that job, did they also replace the desiccant bag and flush the system? If there truly was a problem with the old compressor, they better have. You most likely do not need a fan replaced.