What do you see when you right-click on a Tile on the Windows 8.1 Start Screen? What options, if any, are presented to you at the bottom of the screen?
What happens if you test with another Mouse?
What happens if you test with another User? Create one if needs be.
If you left-click on a Tile and drag it around the screen does it Move for you?
Rearranging tiles on Start
http://windows.microsoft.com/en-AU/windows-8/rearrange-tiles-start
[Windows 8 Start Screen] How to Pin, Resize, Move, Close or Uninstall App Tiles?
http://www.askvg.com/windows-8-start-screen-how-to-pin-unpin-resize-move-and-search-tiles/
How to Resize Start Screen App Tiles to be Smaller or Larger in Windows 8 and 8.1
http://www.eightforums.com/tutorials/4874-start-screen-app-tiles-resize-smaller-larger.html
See if the program can diagnose any problems and fix them.
Download and run the Modern UI App troubleshooter and check.
http://download.microsoft.com/download/F/2/4/F24D0C03-4181-4E5B-A23B-5C3A6B5974E3/apps.diagcab