Bethany, AMICA fares are limited discounts based on date, time and train category. It's simple. You get a 20% discount if purchased before midnight before the day of travel. As you might expect, the Italians are very smart and jump on these limited discounts right away. Still, many posters report that they are often available when they arrive in Italy and purchase tickets days before travel. It is most reasonable to expect that they are not always available when you get there.
Can you purchase AMICA fares online in advance? YES. But only on www.trentialia.com. However, I've been hearing that Trenitalia may no longer be accepting American credit cards beause of too much fraud. I tried booking a ticket a couple of days ago on a credit card that I have used before and it was rejected. I need to call them and see what's up.
On night trains, you might not see the AMICA fares but you do see the SMART and SMART2 fares. Again, these are limited special discounts. But again, the sharp Italian travelers snap these up quickly after they come online (usually 60 days out).
You can see all of these fares and schedules for Italian train travel on www.trenitalia.com. You can buy Italy train tickets much easier on www.italiarail.com but they are a broker and I don't see them passing on the special AMICA and SMART discount fares.