"I suggested he not do it" but it sounds a little formal. I would say "i told him not to do it"

I think that the best way to say it would be: "I suggested/told him not to do it" You may also use reported speech: "I told him he shouldn't do it" Let me know if you have any follow up questions.

#3 is the best. It uses subjunctive mood (by using the infinitive "do"), and that always sounds classy. You can also add "that" to all three sentences. All your sentences are correct. In fact, you can add ANY sentence after an optional "that", and you will have a correct sentence: "I suggested (that) the first man to land on the moon was Neil Armstrong." "I suggested (that) the square root of 16 is 4." It would be strange to say "I suggest that the square root of 16 be 4" although it is a good sentence. By using "be" you employ the subjunctive mood to create an imagined situation. But it is not an imagined situation. Four really is the square root of 16 and that can't be changed, so it's weird. However, if you are with a group of people who are trying to decide what is the square root of 16, it makes sense to say "I suggest that the square root of 16 is 4". #1 is your second best choice. It does not use subjunctive mood but the modal verb serves the same purpose. #2 is correct because "he doesn't do it" is a correct sentence, and it is correct to use ANY correct sentence. However, it has a completely different meaning from #1 and #3. #2 simply states a fact, similar to "I suggested that Molly does not speak German."

I suggested he not do it

I suggested he not do it