Will I lose third party apps if I delete user 501?
Probably not. If you look at your Macintosh HD You should see...

Applications/
Library/
System/
Users/


Typically Applications are installed globally in this top level Applications directory here.

If you look in 

Users/
    |_ MyUserName
        |_ Applications
        |_ Desktop
        |_ Documents 
        |_ Downloads
        |_ Etc Etc.


Hardly ever are applications installed in this Applications directory. 

So unless you see any there. It should be ok to remove the problematic user.