Can the desktop menu bar be hidden?

You can't hide menu bar without using other apps. See > http://osxdaily.com/2011/02/24/hide-mac-menu-bar/ Note that when you are using a full-screen application, menu bar is hidden and it appears when you move the cursor to the top of your screen

Yes, but not using OS X tools. You will need a third-party utility for that.

Here's an article that describes how to get the menu bar behaviour that the OP wanted (i.e., hiding until mouse-over) on an application-by-application basis. It requires nothing more than Xcode to be installed (which is free from Apple in the App Store) and a bit of Property List editing (which is what you need Xcode for).

http://lifehacker.com/5280456/hide-your-mac-menu-bar-and-dock-for-a-cleaner-desk top

The article's a little old, but I just tested it with TeXShop in Mavericks and the method worked perfectly.

Explains everything: http://www.idownloadblog.com/2015/06/22/hide-menu-bar-os-x-el-capitan/

>System Preferences>General