Synchronized means everything is up to date - synchronizing would tell you that it is catching up and may or may not be an issue.
To answer some of your questions:
Primary as backup is ok - unless you license the secondary node then performing backups on that node is not supported and I don't believe a BAG would allow that anyways.
The backups will not affect the status - however, other maintenance can and will have an impact. For example, rebuilding an index will need to be transferred to the secondary and could cause the send queue and redo queues to back up. This is normal but something you should be aware of...
The easiest way to check is to look at the dashboard (you have to add columns to the report to see the send queue, redo queue and other related fields). The report would identify any issues with any databases in the availability group.
A listener is not required - but the listener allows your applications to connect to the node currently hosting the database. Without a listener you have to modify your connections when you fail over.
As for your issue with the backup - that depends on how you have implemented them. Agent jobs do not automatically get created or moved to the secondary, so if the database is on the secondary node those agent jobs on the primary node will not execute successfully.