Not able to Upload Media (Kinda) + Permalinks

A few days ago I wrote in to my host that I was attempting to create a sub-domain (helpdesk.cityofhanahan.com) and that it wasn't functioning normally. The VPS host did something to correct the issue (they stated it was permissions). Now, I seem to have a few problems.

Yesterday, one member of our web team was attempting to refresh some pages and upload new images. It was giving an error while attempting to upload the image from within the page/post. So, I tried to duplicate the problem in the Dashboard >> add new media. The image DOES upload (verified through FTP), but you cannot see it as a thumbnail within WP Media or anywhere else. Further, you cannot attach any of those images to a post/page. While attempting to access the attachment page within WP, it states that it is not found.

I got to thinking and decided to re-select the permalink structure, maybe something was off. This has fixed a problem for me in the past. Come to find out, my "Permalink" option within Settings has DISAPPEARED!

Does anyone have any idea on what is going on, or how to fix this? I have NOT done a single thing to cause the issue and not really sure where to go from here. Host keeps trying different things, but nothing is working.

See attached screen shots.