More about my environment. I have two machines I work from: one at the office and one at home. Both are running Windows 10 Enterprise with VS 2015 Enterprise. I am using visualstudio.com for TFVC. The project has multiple master files, user controls, and pages.
After deleting the .licx file yesterday, removing the Telerik VS Extensions, and redoing the DLLs and references, everything was working fine. I checked in my changes and went home. No new .licx file was created. There is a lib\RCAJAX\2015.2.826.45 folder structure in the solution, but it has no files. I did some work on the project from home and everything was still fine. Checked in my changes and came back to work today. Manually got latest, started working on an ASCX page, and voila, the Telerik entries were automatically deleted from the designer file again.
The VS Get Everything when a solution or project is opened isn't the issue. I don't have that turned on, and the problem isn't that the designer files aren't getting updated, it's that they are getting updated, albeit incorrectly, as soon as I save changes to an ASPX or ASCX file.
It very well could have something to do with version control or with licensing, but without knowing exactly what is happening, I wouldn't want to list all of the things we are trying in the GitHub repo.
The empty lib folder structure is interesting to me. I'm going to try to get this all working again today and see if that has anything to do with it...