AlwaysOn - Running SQL Agent Jobs on Secondary read-only node writing audit information to database
No, it won't automatically write to the primary database instance. You need to connect to the primary instance and perform your data inserts on that instance (linked servers is one option to do this).

You also need to make sure you're running your jobs in the right location. For example, Transaction Log Backups can be offloaded to a secondary, but even a synchronous secondary can run behind the primary (sometimes by a lot of time). If that happens, are you going to be able to meet your RPO goals?

Index maintenance is going to have to run on the primary as it needs a writable database - you can't offload this task.

CHECKDB can be run on the secondary, but I think it should be run on all instances if at all possible. When replication happens it only transfers the log data, not data pages. If you have a disk subsystem corruption on one of your replicas that you don't run CHECKDB on then you may not find corruption until you failover.