Do not is present simple and in narrative sentences is supposed to mean a constant and regular action or some mental process or state to which continuous tense is not applied. It is kind of out of place in this context. Please refer, say, to Wiki for more information or just google "present simple vs present continuous".
Now about the rest of options.
If you have already decided you do not want to show "this" in the report and want to underline this is your plan you are going to follow, you can say:
[..] we are not showing this in the report.
Or, if you are not quite sure yet but are inclined not to show:
[..] we are not going to show this in the report.
Will not has a slight emphasis that the solution has just been done by you. You decide not to show this and immediately say:
[...] we will not show [...]
however, it also mean just the mere fact we will not show this in the report. Refer to this article about differences between be doing, be going to do and will do.
As for need not and should not, you correctly described their meaning, which one to use is up to you and depends on what you want to say.
PS For further reading, you may want to learn more about the phrase to be going to, modal verbs like should, owe, must etc, as well as aspects of present continuous use with regard to the future actions.
Not really. Not only . . . but also assumes parallel subjects.
The best way to connect them, in the example you gave, is simply to drop the also:
Not only should you cook your meals, but they should be healthy.
Alternatively, you could rewrite it to make the subjects parallel:
Not only should your meals be cooked by you, but they should also be healthy.
There you're getting into the passive voice, though, which a lot of people like to avoid. Personally, I would avoid it here, simply because it's too wordy.
One more possibility, again trying to make the subjects parallel:
Not only should you cook your meals, but you should make them healthy.
Slight change in meaning, but works better.
Best Answer
Looking at the first part of this sentence without the not only,
There is something missing from this sentence: you need to use a conjunction that to link the two clauses. You have used should in the second part, and I recommend that you use it instead of do in the first part as well. You don't need a definite article before communicative competence, because competence is normally a non-count noun.
Adding the second part:
Note that not only and but also act as coordinating conjunctions for a parallel structure: They section following them must be the same for both parts. In this case, they appear before the verb in both parts, and students should applies to both of them.
I would also recommend rewording it a little to make it more natural:
You might also consider replace different things by more.