Indexes are backed up as part of the full database backup.
One reason why the database will be running slower is because the live database has most of the database in cache already, while the server you are restoring to does not have any of the data in cache. All the data needs to be loaded off disk for each query until all the needed data is cached in RAM.
Depending on the system configuration of your live and backup database servers the slow response time of the backup system could be normal.
Does the performance ever increase to normal? What’s the hardware configuration for each server?