1. He made a promise with her.

2. He made a word with her.

3. He kept his promise with her.
4. He kept his word with her.

5. He made/kept/broke an appointment with her.
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Can we use all the words such as promise, word, appointment in the sentences?

All instances of "with" in 1-4 need to be changed to "to."

5 is fine.