Krazy is right about ～なくて being simply the ～て form of negative verbs. I think of a compound verb as one that combines a stem form with another verb, like 取り替える. Otherwise we're just linking verbs together.
Also, remember that ～なくて and ～ないで are different; the first is used to link verbs, like 走らなくて困る is "It doesn't run and I'm in trouble," whereas 読まないで分かる is more like "I understand it without reading it" The other use of ～ないで you probably already know is to use it for a negative request 写真を撮らないでください。