Whether you achieve the goal is speculative. If you have already tried to achieve the goal when you were younger (consistent workout over at least a year) and failed it might be indication that it is not possible for you, but not necessarily so.
As for the training, following advice:
Running (2 mile run in 11:54 or less)
Definitely possible in general. In your particular case, we would need to know your current 2 mile, 5 mile and maybe 1/2 mile times. As for the training, I would skip the elliptical machine. It is often argued (see for example, here) that the best way to improve your running is by running. Instead, do 50 minutes of slow and comfortable running once a week. This will have the same effect on your cardiovascular system as the elliptical machine but also get you used to the movement patterns. It is important that you do not worry/plan increasing speed or duration here too much because it will come very naturally soon. In addition, and instead of
Starting at minute and thirty seconds, increasing duration by 3
seconds per day
I recommend doing one interval day per week. Say 8 times running 400m at 6 minutes/mile (if you can already do that, otherwise start slower) with slow running breaks of a minute in between. Every other week, do 4 intervals a 800m (a bit slower). If you can tell me your current 400/800m interval times, I will edit the post.
If you can add a third running day, it would obviously help but that's something you personally have to decide. In case, you have the time go for a second slow 50 minute run.
Push-ups and Sit-ups
Seems generally possible but something most people would not want to achieve because after a time you only train the endurance of your muscles. If you can already do 23 push-ups and 33 sit-ups, it will only be a matter of increasing the number of repetitions. Your plan seems reasonable here. If you get to a plateau after a while try adding some weights and do lower repetitions once in a while. (Still, I am not an expert for high-repetition of push-ups or sit-ups, so let's see what others can add to that.)
Chin-ups
You did not provide any goal for the chin-ups. Please do so, and I will update my post. In any case, be aware that adding one chin-up every week should work very well in the beginning but the plateau effect comes much earlier than for push-ups or sit-ups. Anything above 10/15 seems like a reasonable goal, above 20 seems rather unrealistic.
