If someone asks you the question “Does rebuild index update statistics?”, you probably will say “of course”. You may be surprised to know that index rebuild doesn't update all statistics. Note that non-index stats means the statistics associated with a column/columns that are automatically created or manually created.

Do I need to update statistics after creating index?

You should not update statistics after you rebuilt the index as rebuilding it gives you up-to-date statistics.

Does restore removes index fragmentation or updates statistics etc?

Index fragmentation won't be effected in any way, the restore process restore the data pages exactly 1:1 as before; but file fragmentation cause by file growth will be reduced, because the restore creates the file in one piece.