Live update index in production database
I mean But during index creation/updating period, will they cause
  data corruption because old index cannot be found


None of the documented T-SQL commands can cause data corruption.

The difference between online and offline index operations is in the fact of index's availability: while you do them online, indexes are still available, and when offline, they are not.

This means that your queries just cannot use nonclustered index for a period of its rebuild offline, and in case of clustered index rebuild the whole table is not available when doing it offline.

Online index operations come at a cost: they are slower, they are fully logged (while offline rebuild can be minimally logged if your db's recovery model is simple/bulk logged), and online index rebuild can result in long time blocking under some conditions described here: A SQL Server DBA myth a day: (8/30) unicorns, rainbows, and online index operations by Paul randal