Does the US Constitution apply only to citizens?
No, the rights of both citizens and non-citizens are protected by the US Constitution. However, there are some right specifically reserved for citizens, such as the right to a Federal job and to vote.
In actuality, the Constitution doesn't apply to "citizens," nor does it even apply to "people." It applies to the government. It tells the government what it can and can't do (the body tells the government what it can do, and the Bill of Rights tells it what it can't do).
Immigration rules are administrative ones, and are mandated by Congress, not the courts.
While what is said above is legally true, in reality, non-citizen's rights are NOT protected by the US Constitution. The government cannot completely remove the right of aliens to keep and bear arms, have freedom of expression, etc, but it can greatly restrict those rights almost to the point where they are non-existent. This just goes to show that the interpretation of the US Constitution and the Bill of Rights is really completely arbitary on the part of the government.