Learn English – How to use “not” and “don’t”

‘An A-Z of English Grammar and Usage’ by Leech and others suggests the following:

With be, use the contraction + not (e.g. That’s not right).

With have and modal auxiliaries, use the verb + n’t option, e.g. hasn’t, can’t.

The authors go on to say that forms such as She isn’t hungry, as opposed to She’s not hungry, are less common, and that forms such as I’ve not met him, as opposed to I haven’t met him, are much less common.

The authors don’t say on what basis they make their recommendations, but they are likely to be based on frequency of use. There doesn’t seem to be any difference of meaning or emphasis. I would just add myself that a contraction such as I’ve not met him sounds more formal, perhaps because of its lower frequency.

In your first pair

• Let's not is grammatically "correct" and common in all registers; using it in conversation will not make you sound affected or pedantic.
• Let's don't is grammatically "incorrect" but often heard informally; it should not be used in formal writing, but will not expose you to scorn if used in conversation.

In your second pair

• Make us not practice is grammatically acceptable, but would be used only in conversation; and even there something like keep us from practicing would be more usual.
• Make us don't practice is unacceptable in any register.