Re: differences between letter of application and covering letter?
A cover letter is used to send something else. It could be a resume or a box of samples or anything else, but it's point is to draw your attention to what is being sent.
A letter of application is specifically to apply for something. You may or may not include your resume (if it's a job) or your school records (if it's admission to a university).
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.