May I know which application you are using on your PC or Mac. Try Acrobat or Reader on your machines (PC/Mac) and enable Mobile Link feature. Now when you to your iPhone 6 Plus please login to Adobe Acrobat on it with the same ID and try to access those PDF files.
On the main PC that I am using, I have Adobe Acrobat 9 Pro and thanks to Adobe CC (Student Version) Adobe Acrobat Distiller DC.
The MAC computers at school, I believe that they are using Adobe Acrobat Distiller DC.
I did follow your steps that you recommended, and ensured that the Mobile Link Feature was On. When I go in to the Adobe Acrobat app on my iPhone 6 Plus I am able to see four active locations where my files are located; Recent, Local, Document Cloud, Creative Cloud, and an inactive link for Dropbox (which I don't have an account with). On the Recent location I am able to see PDF documents that I've accessed on my PC recently. Inside my Local location I have the "Getting Started.pdf" file that came with the Acrobat App for the iPhone 6 Plus. In the Document Cloud location I see two files.
What is the Document Cloud? Is the Document Cloud a different service than Adobe Creative Cloud, and if so, where are the files stored and how can one access these files via the web?
Inside the Creative Cloud location I have a sub-folder called "Screenshots," and I can only think that this sub-folder was created when I activated the Mobile Link Feature on the iPhone 6 Plus. Am I correct with this? I also see the actual documents that I know that I have purposely saved in the Creative Cloud file folder.
When I tried to go in to the Document Cloud location to select two PDF files to try and move them on to the Creative Cloud location I get another screen with three choices on the top Cancel, the title of the location, which is "Document Cloud," and Move.
Nothing else appears below, or no other options are given. The only two choices are to either Cancel or to Move (See the screenshots). When Move is selected the Adobe Acrobat will return me back to the Document Cloud Menu, but nothing gets moved.
Adobe Document Cloud is a separate service than Creative Cloud. Find more details here: FAQ | Adobe Document Cloud.
If you have Mobile Link enabled with Acrobat Pro DC and accessing files on your phone that are coming with a cloud icon next to them than it is using Document Cloud. You can also access DC cloud storage by going to https://cloud.acrobat.com/recent. You can move Document Cloud files within sub folders of it.
If you want to move files from DC storage to local storage than use web interface of Document Cloud. More details here: Manage and work with files in Adobe Document Cloud