Outlook has a fairly comprehensive event system. In fact, there are so many events that sometimes it may not be obvious what events will fire when.
Attached here you will find a sample add-in that can show you many of the events generated by an Inspector, Explorer, and a Folder. When monitoring a folder, you will also see Items events. To monitor a folder, right-click it and you'll see a menu option Monitor for events. Choose this, and any folder events that occur in that folder (or that folder's Items collection) will be shown in the event log window. If you monitor two folders, and then move an item from one to the other, you would see the following (in this case the source folder is Test 1, and the target folder is Test 2):
You can monitor as many folders as you like. The inspector events are caught using inspector wrappers (much of the code was based on this sample for Outlook 2010).
Note that this add-in was developed using Visual Studio 2013 against Outlook 2016. It will work with Outlook 2013 as well, though may need to be recompiled against that version (I haven't tested). You can download the zipped solution from here.