[verb ない-form] + で + [verb] [verb ない-form] + で(ください)
~ないで means "without doing": you do the second thing and leave the first undone. Take the ない-form of the verb and add で.
The tense comes at the end of the sentence. ~ないで itself never changes.
~ないで also means "instead of": one choice is dropped and the other taken.
Both are the て-form of ない, and learners swap them. ないで goes with "without" and "instead of". なくて gives a reason: "because it didn't".
With ください after it, ~ないで makes a request not to do something. Among friends, ください drops off.