Should an AG Listener be set up on it's own server? If so, does it require it's own SQL Server license?
The listener is part of the Availability Group and will reside in the resource group of the Availability Group in the Windows Server Failover Cluster. Thus it is not possible to have a specific listener server as the listener will move with the owner of the resources in WSFC.
A listener is composed of hostname and IP address cluster resources that are created in the cluster role that is created for the availability group. So asking if it should be set up "on it's own server" is an invalid question as it is set up as a cluster resource, which could be owned by any node of the cluster that is a replica in the availability group.
The listener itself does not require a SQL Server license, but the instance of SQL Server that is running as the primary of the availability group will require a license. In general, questions about licensing here are not good questions because A) Many licensing scenarios are complex and require research, and B) licensing often changes from one version to the next. So please check with your reseller regarding your license requirements.