Copying applications from applications folder to external and reinstalling osx
Sure. That's definitely possible. I would recommend you use Time Machine to create a backup of your computer, that way you wouldn't lose any data. Also, because you're using 'Apple's way', its guaranteed to work better
  
  
Backing it up: Time Machine
Time Machine, by default, will backup your entire computer. However, you can choose to exclude certain folders from the backup:

In Time Machine preferences you can click the Options button to adjust settings. A sheet similar to this appears when you click Options:

This sheet allows you to exclude files, folders, or entire volumes from being backed up. You might want to do this to avoid filling up your backup drive.

  
  
Restoring it: Migration Assistant
Then, using OS X Recovery, you can do an Erase and Install. Once that is complete, you can use Migration Assistant to bring across certain information, like just your Apps, or perhaps your user information as well.

This would be the preferred way as it ensures just in case, you wont lose anything you explicitly want to and Apple system applications are reinstalled correctly.
  
  
Resources:

HT1427 Mac Basics: Time Machine
HT4718 OS X: About OS X Recovery
PH11273 OS X Mountain Lion: Erase and reinstall OS X
HT4889 How to use Migration Assistant to transfer files from another Mac