> If I have shell script which has a command that runs in the background, is it
> possible to get its return code.

Use wait:

$ sh -c 'false & wait $!; echo $?'
1
$ sh -c 'true & wait $!; echo $?'
0

$ echo aa bb cc dd & wait $! ; echo $?
aa bb cc dd
[1] 13529
[1]+  Done                    echo aa bb cc dd
0

Ref:

$ echo aa bb cc dd & ; wait $! ; echo $?
bash: syntax error near unexpected token `;'

documented on: 22:16:19