Borders exist for a reason - they reflect differences in culture and values. Mexico, for example, is overwhelmingly Catholic. Canada, as another example, does not have guarantees of free speech or religion, and the Canadian people seem largely ok with this. I am not Catholic, and I enjoy the Constitutional protections my rights are afforded by the First Amendment. I would oppose millions of people from either country being allowed to come here and vote without any sort of barrier simply because people's values influence their voting.
In any case, you're creating a false dichotomy; your drunk driving analogy falls flat when you realize that in order for it to work, immigration on its own must be illegal. This is not the case. There are avenues from people all over the world from hundreds of countries to immigrate to the United States.
Wrong. If barriers and borders are most important for a sovereign state, the state enforces the laws that create barriers to citizenship/permanent status, not just "make them legal." Just "making them legal" makes your barriers and borders irrelevant because you've abandoned your protection of them.
And yes, you can say "they can't come in because I decided they can't come in" if you're a sovereign state. Just like I can say that people can't come into my home because I've decided they can't come in. That's the point of sovereignty. I don't need a logical reason, neither does the state. Not all laws are based on hard and fast logic, and neither the Constitution nor the courts say they need to be.