Yes, when you create a clustered index on a heap the rows are all sorted moved to the new clustered index. And any non-clustered indexes are rebuilt with the new clustered index key as the row locator.
It's the same table, though, so triggers, constraints, etc don't have to change.
The reverse is not true, however. When you drop a clustered index on a table, the leaf-level rows of the clustered index are left in-place and they become the new heap.