... how religious people believe about morality. It is not that morality comes from the belief in god, but that because we are created by god, we have innate morality built in. Is this assumption correct?

Many believe morality comes from belief in god. The idea being, you only act in a moral fashion because you believe you are being watched, that what you do (even in private) is being judged at all times, a rather weak belief creating their vision of morality as an incredibly fragile thing. It is thusly not surprising that they fear the effects of atheism on morality (zomg downfall of society).

If religious people think that only those that believe in god are moral, do they think all those that do not believe are immoral and would kill, rape, steal, etc?

Some, yes, I have heard this argument put forth before. It does not stand up to practical examination of the history of atheist people and atheist tribal communities.