Does the train run from the domestic terminal to the international terminal?

Train goes to every Concourse which is every terminal so yes. T-D and partially E are domestic and Some of E and F are international

The train takes you to the concourses on the domestic side. I think it does go to international. I think that you have to go through international security check too. I know that there is also a free shuttle that the airport provides. Just ask an employee which way is best. It takes about 20 minutes. It leaves back to back.

Yes

If your airline has a check in desk at Domestic, you can check in, go through security and ride the train to all concourses. If not, you have to go to International to enter. There is a shuttle bus between Domestic and international. All arrivals that go through customs exit international unless they have a connection.