The second part of your post is something that could and should be easily preventable.
But it is not prevented. Why? The law of large numbers. We have been so inundated by illegal "visitors", we can no longer control it.
I have nothing against immigration. My Great Grandfather immigrated from Germany. Legally. The government knew he was here and where he lived. And he worked hard, jumped through the hoops and became a citizen.
We cannot manage the populace until we manage the ingress of aliens. We need to know who enters the country, where they are and what they are doing. Green cards! Failure to do so is economic suicide and could be literal suicide if some really bad hombres with bad attitudes and nuclear material are let in.
And Trump's wall isn't a solution. Our neighbors to the south are exceptional tunnel engineers. I live 13 miles from the international border and the authorities shut down tunnels in Nogales regularly. Complete with air ventilation, lights and rails to deliver contraband. They would simply go under Donnie's wall.
But we cannot control ingress and movement of immigrants unless we first control our borders. Failure to do so will result in total abrogation of the rule of law.
If someone wants to offer a proposal to make it easier to enter the country legally, I would be willing to listen. But they still must be vetted.
"I swear, by my life and my love of it, that I will never live for the sake of another man, nor ask another man to live for mine."