Yes, last spring I found La Rinascente department store near the Duomo open til 10 pm. The famous Galleria Vittorio Emanuele II shopping mall also may be open when you are able to get settled. Window shopping in the fashion district may be great, depending on weather; I think Rick's guidebook gives a route, or possibly it's on this website. It's possible that furniture stores may be open till 7 pm or later, if that interests you. As for LaScala opera house, most performances begin at 8 p.m.; if there are performances on July 5, I can't see them yet online. If you can go, it will be worth it, but of course that could be a tight timeframe. I don't know if the opera house is lit at night, but I would not be surprised. All of these sites are in very close proximity to one another.
Brera Gallery is open till 7:15 pm but that is very tight; check for possible later opening hours though. Brera is a great district for walking and restaurants.
I found public transportation in Milan to be weak compared to other major European cities. So I would investigate either a taxi or look for tours through Trip Advisor (but book with actual tour company) if you want to be driven around.
I loved Milan, unlike some others who frequent this forum.