I am still not so clear with the difference between '.....しないで.....’ and '....しなくて....' . Could you please explain them for me, and give me some examples?
しないで is an order to not do something. しなくて means "to not do is...".
これをしないで！ don't do that!
これをしなくてもいい。 you don't have to do that.
これをしなくても本当に大丈夫ですか？ is it really ok for you to not do that?
I forgot to mention the following.
しないで can also be used to connect 2 statements in a similar fashion as しなくて but usually used in slightly different ways. I would say that しなくて tends to be more of a cause and effect while しないで to connect rather unrelated statements.
I didn't eat my breakfast and went to school.
I didn't eat breakfast and now I'm hungry.