Yes you can get from Nice to Milan by train... about 5 hours 30 minutes. Most require a change of train at the Italy border but it is very easy. It will be a lovely trip.
If you can ditch your Paris to Nice flights that would be a great idea. The French TGV (High Speed Train) goes directly from central Paris to Nice in less time than it'll take you to get from Central Paris out to the airport, arriving at least an hour before departure and all the other joys of air travel. First Class on the train is about $120 one way and it will be a wonderful experience. The French railway system is nothing like the U.S. system!!!
Here's a link to a train planner
http://www.eurail.com/plan-your-trip/eurail-timetable You won't get anything sensible if you put in dates for next May as a new timetable comes out in December. But, the routes you want are major routes and there WILL be lots of trains for you to choose and plenty of availability.