So if you are trying to find the exit status of the remote command, well, you cant. Also your remsh is setting up a connection with remshd successfully, so it returns success and your "&& echo" command goes through.
What exactly are you trying to do? Do you want to check the exit status of the remote command? I think that you could do it this way:
remsh "$HOST" test -e /home/test.script \&\& echo "yes"
The yes will be printed only if the test result is successful, but it will be printed on the local host, so you can store this in a variable and check the variable for the result. Like this:
result=$(remsh "$HOST" test -e /home/test.script \&\& echo "yes")
if [ "$result" = "yes" ]; then
--remote command succeeded. do something--
else
--remote command failed. do something else--
fi