I'm a Chinese citizen, with an American Green Card. I'm going to travel to Denmark, Sweden, Finland, and Norway in June. Can I apply to Italy for a Schengen visa to travel those countries? I am not going to travel to Italy, but I live in Miami and it's easier for me to go to the Italian Consulate here. I don't want to travel to Houston to apply for the Danish Schengen visa.
No, you must apply at the consulate of your main destination. If you're not traveling to Italy, it cannot be your main destination.
If you can change your itinerary so Italy is your main destination, you can apply at the Italian consulate. If that is not possible, you will have to apply elsewhere.