Prevent ${PIPELINESTATUS[0]} from being evaluated in Jenkinsfile
There seems to be some syntax issues and also I would suggest to use bash instead if using sh shell.

Try the below code and see if this works properly. 
(EDIT - Put shebang on first line to accept as answer)

sh label: 'Stop and Remove Old Docker Container', script: '''#!/usr/bin/env bash
                docker ps -a
                echo $APP_CONTAINER_NAME

                docker ps -a | grep $APP_CONTAINER_NAME || true
                status=("${PIPESTATUS[@]}")
                DOCKERCODE=${status[0]}
                GREPCODE=${status[1]}
                echo "Docker Command Code: $DOCKERCODE"

                echo "Grep Command Code: $GREPCODE"

                if [ $DOCKERCODE -eq 0 ]; then
                    docker stop $APP_CONTAINER_NAME
                    docker rm $APP_CONTAINER_NAME
                else
                    echo "WARNING: Docker command was empty or had an error"
                fi
            '''


In the above code, I have made changes in the way you were trying to get output of command docker ps -a | grep $APP_CONTAINER_NAME || true using PIPESTATUS variable. 
Since PIPESTATUS is a special variable, it's value gets updated after every command, so you were not getting any value of ${PIPESTATUS[1]}

The other change I have made is fixing the syntax issue in the line if [ $DOCKERCODE -eq 0 ] then; to if [ $DOCKERCODE -eq 0 ]; then

You can test out the code further.