I am having a similar issue, none of these enemies are taking damage. i had to basically hope that one of the DoT's would trigger to kill them. However this does not seem to work for the pink/purple creature

Can you explain in detail what you are attempting to do?

As for you, Paganmonk, the monsters in the Wizard arena have elemental shields that only break by taking certain elemental damage. The Holy Wisper can heal them, so it had to be prioritized.

I set the quest as my active quest (the first class mission for the wizard), then I go talk to the Floran and defeat the crazy strong dudes. When I walk out, the quest is still there as if it never completed and just has me reselect it again.

Also those dudes are obnoxious. I got through it after a few tries (still not actually completing the mission) but they are kind of ridiculous for a game like Starbound.

So after you walk out using the teleporter to the right, the quest does not progress to the 'Talk to SAIL' phase? If this is correct, can you send a log file of a time where this happened?

Also, the point is that the mission is hard. With proper gear and a properly leveled Wizard, it should be very doable.