In my state, you go to the circuit clerk, fill out a paper with your name and address, then receive your card in the mail. The receiving the card in the mail is the verification. No ID was necessary for me to register or vote. Seems like this could be gamed fairly easily.

>Do non-citizens even want to vote

Is there something about holding a green card for 5 years that makes you suddenly start to engage in politics?

>How do they even register to vote?

Presumably they take someone else's identity.