What is the proper way to detect shell exit code when errexit option is set?

How about this? If you want the actual exit code …

#!/bin/sh                                                                       
set -e

cat /tmp/doesnotexist && rc=$? || rc=$?                                         
echo exitcode: $rc        

cat /dev/null && rc=$? || rc=$?                                                 
echo exitcode: $rc   

Output:

cat: /tmp/doesnotexist: No such file or directory
exitcode: 1
exitcode: 0