I use a free Select plug-in from Rj-Grafix: http://rj-graffix.com/downloads/plugins/
It allows you to automatically select a ton of different types of objects, including open paths:
In that screenshot, you can also see the “Document Info” window. Open that from Window > Document Info. Click the little hamburger menu and make sure that “Selection Only” is not checked. Then, hit the hamburger menu again and select Objects. It will give you a path count (and if they are open, closed), total path lengths, etc.