It seems you should be able to run solves with upto 32 cores. The cores can be on one or more machines. Please have your system administrator check how many physical cores are available in your machine, you could run with up to those many cores on your computer.
Your cluster administartor should be able to clarify whether your cluster is setup with ARC or a different job scheduler.
if you have access to the license, then you should be able to use it on any machines of your choice. The network administrator should ensure that any firewalls do not block access to license server from these machines.
License will not restrict a machine to the role of headnode or compute, that is cluster layout.
Fluent should be configured to run on all the machines. you may either install onto each mahcine at the same path, or you could install on a file server / application server / headnode and configure the other machines to run the network installation. for details on this please refer the section "188.8.131.52. Network Server and Product Configuration" installation documentation at https://ansyshelp.ansys.com/account/secured?returnurl=/Views/Secured/corp/v232/en/installation/win_install_server.html%23win_prodinstall_server