Can an Italian national marry a green card holder in Texas?

Answer 1

Yes.

As a lawful permanent resident (which is what a "green card" actually represents), you are entitled to most of the same rights and responsibilities as a citizen. Texas (or any other state) has no ability to curtail any of these rights, as they fall under federal jurisdiction.

So, a foreign national alien (i.e. one who does not have a valid permanent US visa) can indeed marry a US permanent resident, and is then eligible to apply for a US permanent visa themselves. However, they will have to wait longer for the paperwork to clear than would a U.S. citizen trying to get a green card for their foreign spouse. Generally speaking, it can take up to a year or so for the green card paperwork to clear.