Accidentally deleted Applications link in Favorites in Finder

It seems you duplicated your applications. To be sure, you can do a Spotlight search of one of your application and see if it's listed twice.

Your "original" applications are under /Applications/.
You can find it by opening Finder, ⌘+⇧+G, entering /Applications in the text box, and validating. It's a folder at the root of your hard drive that Apple put there to hold all the applications for all the users.

Once you checked they're in double, you can delete the ones that are in your user folder.

Your idea of putting all your application under your user account is not a good. Some applications, especially the built in ones, don't like to be moved, and since they already have that special place for them, what's the point?

But, you can do it for your own applications, such as Firefox, Adium, etc.
Those applications can be moved from /Applications/ to the Applications folder in your user account.

Final question, why do you want your application listed under your user Applications folder?

I you really want to do that, I guess you can make an alias.

Go into /Applications/, select all the applications, hold ⌘+⌥ while moving them into the new location.
There's a little arrow under that will tell you that they will only be aliased:

Drag the folder to the desired position in the sidebar. This adds the folder to the sidebar in every current and future Finder window, as well as Finder Open/Save windows/sheets.

Removal is done by dragging the folder from the sidebar out to the side.