- It's important [that you do not be late]. -- [? ? ?]
- It's important [that you not be late]. -- [subjunctive mandative]
- It's important [that you are not late]. -- [covert mandative]
Your last two examples (#2, #3) are unremarkable, and are completely acceptable in today's standard English. It's that example #1 that is interesting, and I'm assuming that is the one that you are mainly interested in discussing. Here are my thoughts on what the subordinate clause in example #1 might be.
Example #1's subordinate clause doesn't seem be an imperative clause for several reasons. (Though, the sentence does seem to be intended by the speaker to be a mandative construction.)
First reason: The analytic "do not" is of somewhat doubtful acceptability in (main clause) imperatives that have "you" as overt subject, e.g. "You do not be so cheeky". (Info borrowed from CGEL page 928-9.)
CGEL says on page 802-3 that:
- "[ Analytic do not ] is particularly unlikely when the imperative has an overt subject, and in the case of you it can be regarded as ungrammatical: …"
and the ungrammatical example they provide is:
- [9.ii.b] * Do not you renege on our deal.
Second reason: In imperative clauses, the versions that have an overt 2nd person subject ("you") would have the same meaning as those that have no subject and those that use a vocative. For instance:
- a. "Come here." -- [no subject]
- b. "You come here." -- [subject]
- c. "You, come here." -- [vocative]
Those above three examples are basically interchangeable, especially as to their meaning; and also, I would expect all three versions to have the same level of acceptability.
But in example #1, its subordinate clause doesn't seem to have the same acceptability (nor, perhaps, even the same meaning) for the corresponding alternates:
- 1.a It's important [that do not be late]. -- [no subject]
- 1.b It's important [that you do not be late]. -- [original #1, subject]
- 1.c It's important [that, you, do not be late]. -- [vocative]
Version #1.a seems to be completely unacceptable; and version #1.c seems weird--though, its subordinate clause does sound like it is an imperative clause. And so, the three subordinate clauses don't seem to be interchangeable here, and that gives more weight to not consider the original example #1's subordinate clause to be an imperative clause.
CAVEAT: Though, it probably should be pointed out that a main clause imperative in a coordination might not always do so well here either. (Examples of imperatives in a coordination: Say what you like, it won't make any difference; You do that again and you'll regret it; Do that ever again and I'll brain you; Act in haste and repent at leisure; Come over here and you'll be able to see better; You hurry up or we'll be late.) But from those examples, the only one that I'm seeing as being awkward is "You come over here and you'll be able to see better".
What it might be: Example #1 might be an attempt by the speaker to combine an emphatic DO with a subjunctive clause. That is, the "do" in "It's important [that you do not be late]" might be an emphatic "do"; though, it does seem to be unnecessary.
I'm thinking this because example #1 seems to convey the same meaning as the (subjunctive) mandative. Example #1 might look a little bit unusual, but otherwise, it doesn't sound too bad to me (AmE speaker).
Now, for your last question, which concerns the following example:
- It's important [that you don't be late].
This example is basically the same as your example #1, except that this example is using synthetic negation instead of analytic negation. Let's see how the three alternate versions ("you" as overt subject vs no subject vs vocative) look:
- 4.a It's important [that don't be late]. -- [no subject]
- 4.b It's important [that you don't be late]. -- [original #4, subject]
- 4.c It's important [that, you, don't be late]. -- [vocative]
Version #4.a seems to be completely unacceptable; and version #4.c seems weird--though, its subordinate clause does sound like it is an imperative clause. And so, the three subordinate clauses don't seem to be interchangeable here. This is the same conclusion we had when we performed this test with example #1a-c.
The main difference between example #4 and your original example #1 is that example #4 seems to sound smoother (to me).
NOTE: CGEL is the 2002 reference grammar by Huddleston and Pullum et al., The Cambridge Grammar of the English Language.