I am relatively new to creating process maps. Is it common to map out exceptions to a "happy path" process (handling errors), or is there a better way to document those exceptions?
A common approach is to use Process Failure Modes and Effects Analysis in conjunction with a happy path process map.
The purpose of P-FMEA is to identify potential ways each step in the process may fail, the impact of failure, and the ability to detect that failure has occurred. Once a P-FMEA analysis has been conducted for a process, then the list of possible failure modes is prioritized and there are three courses of action for each item in the list: defer, redesign, or mitigation. Handling of low priority failure modes can be deferred until higher priority failure modes are addressed. The process can be redesigned to eliminate the possibility that a failure mode occurs. Ability to detect occurrence of a failure mode can be improved, along with establishing a response plan when that mode is detected. See FMEA on Wikipedia for a more detailed introduction and additional information about Process FMEA.
Your process map is not complete unless you map out every possible path work can flow, no matter how small its frequency or probability. Otherwise, when that very small possibility occurs, your process will fail and no one will know what to do.
In real life, however, you may not have the time to map it out, design it, or even identify where an exception could occur because your process map is likely an input to the next piece of work for whatever transformation you are going through. So this typically becomes an action to resolve as your overall all capabilities under the transformation effort matures. Building upon your process map, both to make it more efficient and of higher value as well as to map out low occurring paths, is a maturity thing. Takes time.