I use Outlook 2013 with Exchange Server and on both my desktop and laptop the Outlook subfolders of the Inbox have recently started to act weird. They will collapse at random sometimes quickly but most of the times one by one. Since I have several hundred subfolders this process takes a couple of minutes. There seems to be little logic to this happening, other than that it is happening more and more frequent. Sometimes it happens when I click on a sub-folder, 3 times in a row. Then the 4th time the sub-folder will open as normal. Sometimes it happens when I resize a window or click the mouse anywhere for that matter.

Any idea what's going on here?

Probably some add-in acting funny, or additional software installed, especially mouse-related. Try running Outlook in safe mode for a while, see if the issue reappears. To launch Outlook in safe mode, hold the CTRL key when clicking the icon or type "Outlook /safe" in the run box.