WesT's answer points out that expanding the contraction works if you allow the subject to be omitted. In fact, rather than don't being an unusual feature of the examples in the question, it's the you that was the unusual feature; it's more usual with the English imperative to omit it, though it can be included for emphasis.

Part of the question that wasn't dealt with is:

"Don't" is a bit unique among "not" contractions (at least that I know of) because it is the only one that can be used in an imperative sense. Couldn't, shouldn't, wouldn't, haven't, isn't, doesn't, aren't, can't... none of these can be used in an imperative sense because you cannot command someone to could, should, etc.

It's not so much don't that is unique in this, as do.

Some constructs can only happen in English with an auxiliary, particularly inversion and negation.

Inversion first:

He will make up his mind.

 

Will he make up his mind?

 

He decided.

 

*Decided he?

This last isn't allowed because decided isn't an auxiliary, and so we use do (inflected as did):

Did he decide.

Similarly with negation:

I will eat it.

 

I will not eat it.

 

I eat it.

 

*I eat not it.

Again this is not allowed in English (not in contemporary standard English, anyway), so we use do again:

I do not eat it.

Since the negative imperative is of course a case of negation, if the verb used is not an auxiliary (as it generally won't be, more on that later), then this same "do-support" is needed, and hence "Don't touch that button" is the negative of "Touch that button".

The other auxiliaries tend not to be used with the imperative, not so much because it would be grammatically incorrect, as it would be meaningless. What does it mean to instruct someone with will (in the auxiliary sense), or can?

With might, mote, may (in most senses) or would trying to form an imperative clause finds no reasonable target except perhaps fate or the divine and turns into a wish or a blessing: "May you be happy!", "Would that the rain would stop!". Describing this as imperative is a bit of a stretch!

Some dialects (some in Ireland certainly, and I gather some in India), do use may in the imperative to express a polite command (though the politeness can be sarcastic): "May you please join me tomorrow", "You may reply as soon as possible". Other dialects wouldn't use this and those who speak them perhaps misinterpret these expressions as giving permission rather than request (indeed, precisely where the form comes from originally; giving permission and letting the command be implied).

Outside of that example, do is the only auxiliary that would be meaningful in the imperative.

WesT's answer points out that expanding the contraction works if you allow the subject to be omitted. In fact, rather than don't being an unusual feature of the examples in the question, it's the you that was the unusual feature; it's more usual with the English imperative to omit it, though it can be included for emphasis.

Part of the question that wasn't dealt with is:

"Don't" is a bit unique among "not" contractions (at least that I know of) because it is the only one that can be used in an imperative sense. Couldn't, shouldn't, wouldn't, haven't, isn't, doesn't, aren't, can't... none of these can be used in an imperative sense because you cannot command someone to could, should, etc.

It's not so much don't that is unique in this, as do.

Some constructs can only happen in English with an auxiliary, particularly inversion and negation.

Inversion first:

He will make up his mind.

Will he make up his mind?

He decided.

*Decided he?

This last isn't allowed because decided isn't an auxiliary, and so we use do (inflected as did):

Did he decide.

Similarly with negation:

I will eat it.

I will not eat it.

I eat it.

*I eat not it.

Again this is not allowed in English (not in contemporary standard English, anyway), so we use do again:

I do not eat it.

Since the negative imperative is of course a case of negation, if the verb used is not an auxiliary (as it generally won't be, more on that later), then this same "do-support" is needed, and hence "Don't touch that button" is the negative of "Touch that button".

The other auxiliaries tend not to be used with the imperative, not so much because it would be grammatically incorrect, as it would be meaningless. What does it mean to instruct someone with will (in the auxiliary sense), or can?

With might, mote, may (in most senses) or would trying to form an imperative clause finds no reasonable target except perhaps fate or the divine and turns into a wish or a blessing: "May you be happy!", "Would that the rain would stop!". Describing this as imperative is a bit of a stretch!

Some dialects (some in Ireland certainly, and I gather some in India), do use may in the imperative to express a polite command (though the politeness can be sarcastic): "May you please join me tomorrow", "You may reply as soon as possible". Other dialects wouldn't use this and those who speak them perhaps misinterpret these expressions as giving permission rather than request (indeed, precisely where the form comes from originally; giving permission and letting the command be implied).

Outside of that example, do is the only auxiliary that would be meaningful in the imperative.