Yes, that is correct. It is mostly said in situations where you want to portray deep emotion. It would be odd to use it in business situations, perhaps? Or more formal situations, unless you wanted them to feel that it was really important to you in a very personal way.

Yes. It's perfectly correct. For example, a Google search on site:gutenberg.org "left a deep impression" turns up "I remember, as a boy at Eton, seeing a scene which left a deep impression on me." and "What you said yesterday ... left a deep impression on me."

Yes, that is perfect. You could also use the adjective "profound" in place of "deep". A profound impression.

This is grammatically correct, but "This left a deep impression on me" will sound more natural in most contexts.