なくて

なくて is the て-form of a negative, and here it gives a reason: "because (it) didn't", "because there was no". Take the ない-form and change the final い to くて. The second half says what followed – usually a feeling, or a situation nobody chose.

No requests or plans after it

A なくて reason cannot lead into a request, a suggestion or something you intend to do. Those need から or ので.

• × 時間がなくて、タクシーで行きましょう → 時間がないから、タクシーで行きましょう
• × お金がなくて、貸してください → お金がないので、貸してください

なくて or ないで

Both are て-forms of the negative. なくて gives a reason. ないで says what someone did instead, or did "without" doing.

• × 薬を飲まなくて、学校に行った → 薬を飲まないで、学校に行った

Adjectives and nouns give reasons the same way: ひろくなくて広くなくて, じょうず上手じゃなくて. After a noun, じゃなくて more often means "not X but Y", which is a separate point.