Database maintenance activities are not replicated. If/when you need to rebuild indexes, you must do this on both databases.
As for the performance impact of having indexes on your replica, the short answer is "Yes, any index will have impact on the performance of INSERTs, UPDATEs and DELETEs". The real question is whether the impact on those commands is significantly less than the performance benefit you gain for other queries by having those indexes in place. You may need to test/quantify the impact each index has on your system.