How to check whether a background job is alive? (bash)

You can get the PID of the most recent background job with $!. You could then check the exit status of ps to determine if that particular PID is still in the process list. For example:

sleep 30 &
if ps -p $! >&-; then
    wait $!
else
    jobs
fi

You can check if a signal is deliverable

./script2 &
myPid=$!
sleep 5

if kill -0 "$myPid"; then
    script3 &
    wait
fi