No, SQL Server will not start to move data around just to rebalance - moving lots of data is expensive. But if you rebuild an index, the new version of the index may appear in one of the new files.
As Olaf and Lucy said, SQL Server writes to the files in a round-robin fashion. Therefore, the best if the files have the same size and autogrow settings.
What will happen if you add completely new files I don't know. But it sounds corny, if would insist on doing round-robin over files that are full and cannot grow.
I would recommend that you restore a backup of the database on an idle machine, add the new files, and then add some load on the database to see what happens.