At what stage in academia are you most likely to suffer a mental breakdown? Let's see, just in this week alone I got the news that I have failed four of my midterms (out of five courses in total, the last course doesn't have a midterm), achieving around 40% in each, quite consistent effort across the board. By the way I have around a 3.8 GPA over 4 years of engineering school and I received a letter from the faculty telling me to pull my grade up or consult mental health professional.

I think all this is happening I am in the final stage of wrapping up a year long project that has truly went awry. Two of the members contributed nothing and did not even participate in completing the final system. So I was there to build the entire system myself. This was easy until it was time to deliver our final report. Guess who wrote the 80 pg long final report.

During all this I had one huge McDonald per day at around 1:00 AM, drank a dozen of energy drinks which I know cannot be too good for me. Missed nearly every single lecture. Wrote a 1000 word email to my supervisor detailing how much I suffered through this project and later wrote a 20 word email apologizing for the rambling. Fell sleep in the middle of a lab exercise (thankfully I am able to make up for the lab tomorrow morning). Also rejected from two schools, not my dream school, but hurt nonetheless.

This is the part when I think I would be finally schizophrenic. Like John Nash or that girl in Proof, I go full blown insane. Wear a clown hat on my way to school while swearing at bus drivers, sitting in the middle of a floor in a 7-eleven crying or writing indecipherable symbols on the blackboard. I am not dealing with this stress at all. I have not talked to my professors about the midterms, no plans to make up anything. I have not bothered to pick up the midterms either to check what is wrong.

Surprisingly, none of this has happened. I wonder if I am in a phase that is beyond five stages of grief, or if I have truly given up in my subconscious. Has anyone experienced something similar? Like a setback that is so big that you would think you would run into emotional troubles but nothing wind up happening. At what stage in academia would one most likely to suffer from a break down of some sort?
You have given more than enough evidence to show that you need to seek professional help and/or counseling.  The impulse of telling people that you're in trouble is a good one.  Posting anonymously on a site for academic advice is not the way to go: because you clearly need help by those with professional counseling training and very few academics have such training, most people on this site will (and should) decline to give you specific advice.  Please seek help in real life.  You can begin with student counseling services at your university.  If you don't know where to go, google "student counseling [your university]".  

Note: I am leaving this as an answer rather than a comment because I feel strongly that this is "the real answer" for the OP.  The manner in which probability of mental breakdown varies with the stage of one's academic career is not relevant to the OP.  Mental breakdown can happen at all stages of an academic (and nonacademic) career, and rather than looking at age or seniority statistically one should look behaviorally, cognitively and emotionally on a case-by-case basis.