Welcome to Android Central! What phone are you using? When you talk about the old Messaging app icon going away, are you talking about icons in the Notification Bar when you get a message, or are you talking about the shortcut on the homescreen? If you bring up the App Drawer, can you find the stock Messaging app there? If it's there, you should be able to drag it back to a homescreen.
It's possible to make Hangouts your default SMS app, which may be what you did. Go to Settings>Wireless & Networks>More, and see what's listed under Default SMS App. If it's Hangouts, select the stock Messaging app again. You can also open Hangouts, tap Menu, then Settings, then SMS, and see if it says "SMS Enabled."