(1) Based on your support registration, you are using the Ansys Lumerical Academic Teaching License (25 tasks).
(2) Business or Commerical license users can register for commerical support with Lumerical as shown in our KB: > https://optics.ansys.com/hc/en-us/articles/5609426062483. Where commerical license users can submit/open a ticket in the Lumerical support portal > https://optics.ansys.com/hc/en-us.
(3) Yes, you can run parameter sweeps on Lumerical including MODE (EME/FDE/varFDTD). You should have 25 MODE FDE solver licenses. You can set your resource configuration to run using the available cores on your machine. See the following KB articles for details:
You can run concurrent parametric sweeps on your local PC or remotely as shown here: >https://optics.ansys.com/hc/en-us/articles/360026162414
(4) Ensure that you are accessing the license server with available licenses. Check the license status and availabity as shown in this page: >https://optics.ansys.com/hc/en-us/articles/5770622400659